CONVERTED 


LIBRARY 


1 


UNIVERSITY  OF  CALIFORNIA. 

Received     '^/J'Zt^'ly .  ,  /  8c)6 . 

Accessions  hloP  / ^^^O.  Class  No. 


A  \yj 


OUR   NOTIONS 


OF 


Number  and  Space 


HERBERT    NICHOLS,    Ph.D. 

LATE  IXSTRCCTOR  IN  PSYCHOLOGV,  HARVARD  IXIVERSITY 


ASSISTED    BT 


WILLIAM  E.  PARSONS,  A.B. 


7B^  •ITY)j 


irh^ 


BOSTON,  U.S.A. 
GIXN   &   COMPANY,    PUBLISHERS 

1894 


JV  5 


Copyright,  1894 
BY  HERBEKT  NICHOLS 

ALL  RIGHTS  RESERVED 


CONTENTS. 


Page. 

Introductiox           .........  V 

ExPERi.MEXT  A.  —  With  Pins  set  in  a  Straight  Line   .         .  1 

Experiment  B.  —With  Pins  set  in  Triangles  and  Squares  21 

Experiment  C.  — With  Lineal  Figures        ....  40 

ExPERi.MENT  D.  —  With  S(jlid  Figures  .  .         .43 

Experiment   E.  —  With  Moving  Pencil         ....  57 

Experiment   F.  —  Comparing  Horizontal  ami  Vertical  Dis- 
tances          63 

A  Study  of  the  Kesults    .......  71 

Number  ..........  71 

Distance 86 

Number-Judgments  based  on  Two  Dimensions     .         .  Ill 

Distance-Judgments  based  on  Two  Dimensions    .         .  126 

Judgments  of  Figure 142 

The  ilass,  Intensity  and  Time  Elements  of  Distance- 
Judgments        ........  147 

ExPERi-MENT  G.  — With  a  Single  Pin  .....  156 

Experiment  H. — Education  of  Artificial  Space-Relations.  172 

General  Survey  and  Summary  ......  177 


INTRODUCTION. 


My  thesis  I  briefly  state  as  follows :  Our  brain 
habits,  with  the  modes  of  thought  and  of  judgment 
dependent  thereon,  are  morphological  resultants  of 
definite  past  experiences  :  our  experiences,  and  those  of 
our  ancestors.  Each  limited  experience  does  its  share 
toward  fixing  a  limited  habit.  The  experiences  most 
common  to  our  various  regions  of  skin,  differ  widely 
one  from  another  ;  those  of  the  tongue,  from  those  of 
the  fingers  ;  those  of  the  fingers,  from  those  of  the 
abdomen,  and  so  on.  Our  habits  of  judgment,  based 
on  these  several  avenues  of  experience,  ought  therefore, 
when  compared  with  each  other,  to  betray  permanent 
characteristics  running  parallel  with  the  local  differ- 
ences of  anatomy,  of  function,  and  of  experience,  which 
give  rise  to  them,  and  in  which  they  are  rooted. 

Investigation  proves  this  to  be  the  case.  It  shows 
that  our  judgments  of  the  same  outer  facts,  such  as 
of  number  and  of  distance,  vary  greatly  when  mediated 
by  different  tactual  regions.  And  what  is  of  greater 
importance  to  the  science  of  psychology,  these  varia- 
tions in  judL;ment  bear  distinguishing  ear-marks  of  the 


VI  INTRODUCTION. 

kinds  of  experience  out  of  which,  and  by  reason  of 
which  through  life,  they  have  slowly  risen. 

It  is  our  purpose  to  study  these.  Through  compari- 
son of  the  different  constants  and  variables  in  certain 
judgments,  which  we  shall  subject  to  experimental  proof 
and  analysis,  we  aim  to  discover  somewhat  regarding 
the  fundamental  laws  governing  the  past  genesis  and 
the  present  formation  of  our  judgments,  and  of  the 
movements  of  mental  processes  in  general. 

As  the  last  words  of  this  Introduction,  I  wish  to 
thank  Professor  Miinsterberg  for  permitting  one  of 
his  students  to  assist  me  with  this  research  during  an 
entire  year.  And  with  deep  appreciation,  and  pleasant 
recollections,  I  record  the  patient  labor  and  able  service 
which  Mr.  Parsons  has  continually  contributed  to  the 
work. 


OUR  NOTIONS  OF  NUMBER  AND  SPACE. 


EXPERIMENTS  A,  B,  C,  D,  E,  F,  G,  and  H. 

These  several  experiments  form  a  set.  "We  shall 
first  present  the  method,  and  the  bare  results  of  each 
one  separately,  then  study  them  collectively. 

EXPERIMENT    A. 

WITH    PINS    SET    IX    A    STRAIGHT    LIKE. 

Apparatus.  —  Heavy  cardboard  was  cut  in  strips  7  or 
8  mm.  narrower  than  the  pins  to  be  used.  The  pins 
were  the  familiar  household  article  ;  they  were  run 
through  the  whole  width  of  the  strip,  which  held  them 
firmly,  their  ends  projecting  like  the  teeth  of  a  comb. 

Thirty-six  cards  were  thus  prepared,  or  9  sets  of  4 
cards  each.  The  ''  9  sets "  corresponded  to  the  9 
distances  experimented  with  ;  and  by  "  distance "  we 
shall  always  denote  the  distance  between  the  end  pins 
of  tlie  line  of  pins.  The  9  distances  embraced  the 
even  and  the  half  cmm.  from  1  to  5  inclusive. 

The  4  cards  of  each  "  distance  set "  were  fitted  with 
2,  3,  4,  and  5  pins   respectively.      These   pins  (Avhen 


2  OUK    NOTIONS    OF    NUMBER    AND    SPACE. 

more  than  2)  were  spaced  equally  apart  for  each  card  ; 
these  sub-distances,  of  course,  varying  on  each  separate 
card. 

Thus  prepared,  our  36  cards  represented  9  categories 
of  "distance,"'  and  4  categories  of  "number"  for  each 
distance.  (For  the  abdomen  an  extra  6-pin  card  was 
used.) 

A  holder  was  provided  for  these  cards,  in  order  that 
the  subject,  wlien  taking  them  in  his  hand,  should 
learn  nothing  about  them  through  the  pinch  of  his 
fingers.  This  holder  was  a  folded  strip  of  sheet-steel, 
—  the  cards  were  dropped  into  its  groove,  with  the 
heads  of  the  pins  resting  against  the  metal,  and  the 
sides  of  the  holder  were  then  pinched  together. 

Method.  —  In  this  experiment  the  subject  applied 
the  pins  to  himself,  the  cards  being  drawn,  put  into 
the  holder,  and  handed  to  him  by  some  one  else.  In 
applying  the  cards  the  subject  was  permitted  to 'rock 
the  pins  back  and  forth  on  his  skin.  The  line  of 
direction,  in  which  the  line  of  pins  were  applied,  was 
for  each  locality  always  the  same.  This  was  at  right 
angles  to  the  median  line  on  the  tongue,  the  forehead, 
and  the  abdomen,  and  longitudinally  on  the  forearm. 

At  first  an  instrument  was  used  to  regulate  the 
pressure  with  which  the  pins  were  applied  ;  but  it  was 
soon  found,  the  pins  being  sharp,  that  the  subject's  own 
feeling,  adapting  itself  very  sensitively  to  the  conditions 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.  3 

of  best  judgment  under  varying  conditions  of  thickness 
and  toughness  of  skin,  was  a  better  "  control "  for  the 
"  constancy  of  pressure  "  than  any  mechanical  contriv- 
ance could  possibly  be. 

Proper  care  was  used  to  avoid  complications  due  to 
fatigue  or  to  changes  of  temperature. 

Exjilanatlons  of  the  Tables.  —  In  this  experiment :  A 
four  persons  were  experimented  upon  as  follows.  B, 
a  student  of  biology  ;  L,  a  student  of  psychology  ;  P, 
Mr.  Parsons  ;  and  N,  myself.  Each  card  of  pins  was 
applied  100  times  to  each  person.  The  "  distance " 
categories  are  indicated  in  the  left-hand  vertical  column 
and  govern  the  horizontal  line  of  figures  opposite  to 
them,  across  the  page.  The  "number"  categories, show- 
ing the  number  of  pins  in  each  card,  are  indicated  by 
Roman  numerals  in  the  top  horizontal  heading,  and 
govern  the  vertical  column  of  figures  below,  to  the  bottom 
of  the  page.  The  main  body  of  figures  shows  averages 
calculated  from  100  applications  of  each  card  to  each  of 
the  four  persons,  i.e.,  from  a  total  of  400  applications. 

The  four  main  horizontal  divisions  of  the  tables  show 
as  follow  :  — 

The  First :  —  Shows  the  number  of  times,  per  hun- 
dred times  applied,  that  the  number  of  pins  Avas  judged 
correctly. 

The  Second :  —  ShoAvs  the  per  cent,  error  made  in 
judging  the  number  of  pins. 


4  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

The  Third :  —  Shows  the  number  of  times  per  hundred 
times  applied,  that  the  distance  was  judged  correctly. 

The  Fourth :  —  Shows  the  per  cent,  error  made  in 
judging  the  distance. 

Tables  1,  2,  3,  and  4  are  wholly  ''  regular  "  according 
to  the  foregoing  explanations. 

Table  5.  —  The  question  arose  as  to  what  part  the 
rocking  of  the  pins  back  and  forth  on  the  skin,  which 
was  permitted  the  subject,  played  in  making  his  judg- 
ment. Or  to  put  the  matter  more  psychologically,  the 
question  rose  as  to  how  far  such  judgments  were  direct, 
and  how  far  complex  and  reasoned  out.  To  throw  light 
on  tliis  matter,  a  set  of  tests  was  made  upon  the  fore- 
arm, which  differed  from  the  regular  experiments  in 
that  the  pins  were  only  permitted  to  be  pressed  upon 
the  skin  steadily  and  evenly  throughout  the  whole  line, 
and  but  three  times  in  regular  succession,  at  intervals 
one  second  apart.  The  results  so  obtained  are  reported 
in  Table  5. 

Table  6.  —  Particularly  it  was  suspected,  that  our 
judgments  of  the  number  of  pins  in  a  given  card 
were  reasoned  out  somewhat  as  follows  :  that,  feeling 
perhaps,  the  two  end  pins  widely  apart,  and  two  inter- 
mediate pins  nearer  together,  or  even  one  pin  at  some 
intermediate  point,  we  then  said  :  —  "  since  the  inter- 
mediate pins  are  spaced  equally  there  ought  to  be  so 
many   pins";    thus   arriving  at   the    final   estimate  by 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.  5 

mathematical  calculations   based   on   the   partial    data 
actually  given  in  the  impression. 

It  was  found  that,  Avith  practice,  this  sort  of 
reckoning  process  could  be  largely  suppressed  by 
volition ;  by  concentrating  our  attention  upon  the 
number  of  points  felt,  and  strenuously  shutting  out 
all  else  from  the  impression.  If  we  could  not  suc- 
ceed in  this  perfectly,  it  was  well  to  compare  results 
obtained  by  this  method,  with  those  where,  as  in 
the  regular  experiments,  every  possible  aid  was  given 
to  forming  the  judgments.  Table  G  presents  such 
results. 

Table  7.  —  It  being  a  main  proposition  of  this  re- 
search to  compare  results  obtained  upon  dissimilar 
regions  of  the  body,  it  was  requisite  that  all  the 
results  should  be  obtained  under  conditions  as  similar 
as  possible.  As  a  matter  of  fact  the  different  regions 
were  for  each  separate  experiment  worked  upon  suc- 
cessively, and  the  full  set  of  tests  was  finished  for  one 
region  before  proceeding  to  another.  At  the  end,  the 
question  arose:  how  were  the  later  results  influenced  by 
the  considerable  amount  of  skill  and  practice  acquired 
in  the  foregoing  work?  To  test  this,  still  another  series, 
perfectly  regular  in  its  method,  was  taken  upon  the 
forearm,  at  the  very  end  of  all  our  work.  Table  7 
contains  its  results.  The  regions  were  first  Avorked 
on  in  the  following jo^^err^v^-xtongue,  forehead,  fore- 

[uhivee:it7) 


6  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

arm,  and  abdomen,  and  the  corresponding  tables  are 
arranged  in  similar  order. 

Tables  8  and  9.  —  For  theoretical  reasons  to  be 
reached  in  our  future  discussions,  it  became  desirable 
to  have,  for  comparison  with  our  other  results,  a  set  of 
judgments  from  impressions  where  the  distance  cate- 
gories remained  the  same  as  in  our  regular  experiments, 
but  where  the  number  of  pins,  or  points,  in  each  line 
should  be  increased  to  a  maximum,  or  to  infinity.  That 
is,  wherein  a  straight  line,  or  straight  edge  should  be 
pressed  upon  the  skin,  instead  of  pins  set  at  intervals 
in  a  line.  Accordingly  a  series  of  tests  was  made  upon 
the  forearm,  with  a  set  of  cards,  cut  to  proper  lengths 
from  thin,  hard  card-board,  the  whole  length  of  the 
edge  of  the  card  being  pressed  directly  upon  the  skin. 
Table  8  shows  results  obtained  Avith  them,  according 
to  the  regular  method  of  permitting  the  subject  to 
''rock''  the  card  upon  the  skin.  Table  9  shows  results 
comparative  with  those  of  Table  5,  where  the  cards 
were  pressed  evenly  and  steadily  three  times  in  suc- 
cession, at  intervals  one  second  apart. 

Table  10  is  a  general  summary  of  the  foregoing 
tables,  and  aids  in  comparing  the  different  regions 
worked  upon. 

Note.  —  A  glance  at  any  of  our  tables  shows  them  divided  into 
sub-tables,  or  blocks.  Each  block  bears  a  number,  in  parenthesis, 
in  its  upper  middle  portion.     I  shall  always  refer  to  these  minor 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.  7 

tables  as  "  blocks,"  identifying  them  by  their  proper  numbers. 
The  total  number  of  blocks  is  356.  But,  owing  to  the  great 
expense  that  would  be  incurred,  I  am  unable  to  publish  the 
results  obtained  from  the  individual  subjects,  and  can  lay  before 
the  present  reader  only  the  blocks  and  figures  which  present 
the  averages  calculated  from  the  four  subjects.  The  original 
figures,  however,  are  at  the  service  of  any  one  who  cares  for 
them. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


Table  1.      Bzperiment  A.  —  Pins  in  straight  line. 
TONGUE. 


Subject 

Distance 
between 
End  Pins 

(Centi- 
meters) 

Average  of  N.  P.  L.  ajtd  B. 

Xo.  PiXS  IX 
LiXE 

11             III 

IV 

V 

Averages 

o   5 

(5) 

g 

1  2| 

1 

100 

97 

96 

99 

Ph 

1.5 

100 

99 

98 

99 

o 

'm    o    S- 

2 

100 

100 

97 

98 

a 

e  ^  " 

2.5 

100 

100 

98 

97 

a 

ii 

3 

100 

99 

97 

96 

Averages 

100 

99 

97.3 

97.8 

98.5 

'4a 

O 

a 

5 

(10) 

ir. 

w  d    . 

1 

0 

+  1.1 

+  1.2 

-  .3 

-  ^  "^ 

1.5 

0 

+   .3 

+   .5 

-  .2 

S 

«-  °  s 

•> 

0 

0 

+   .3 

-  .4 

§ 

(^2-^ 

2.5 

0 

0 

+   .5 

-  .2 

o 

3 

0 

+   .3 

-    .2 

-  .8 

Averages 

0 

+   .3 

+   .5 

-  .4 

+  .075 

(15) 

'A 

III 

1 
1.5 

98 

08 

99 
91 

97 
91 

100 
97 

98.5 
94.2 

»s 

«  s3  a. 

2 

95 

94 

95 

97 

95.2 

f- 

IS^ 

2.5 

91 

95 

96 

98 

95.0 

n 

«i 

3 

98 

94 

93 

97 

95.5 

O 

Averages 

96.0 

94.6 

94.4 

97.8 

95.68 

(20) 

-AA    .£ 

1 

+  1.0 

+   .5 

+  1.5 

0 

+    -7 

Q 

go's 

1.5 

+   .7 

+  3.0 

+  2.5 

+  1.3 

+  1.9 

D 

"     ^-f 

2 

+  1.2 

+  1.2 

+   .1 

-   .4 

+   .5 

(So-^ 

2.5 

+  1.4 

+   .4 

+   .6 

+   .2 

+   .6 

3 

-   .3 

-1.0        -1.2 

-   .5 

-  .7 

Averages 

+   .8 

+   .8     1   +   .7 

+   .1 

1*=^ 

10 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


Table  2.      Experiment  A.  —  Pins  in  straight  line. 
FOREHEAD. 


Subject 

Distance 
between 

Average  of  N.  P. 

L.  AND  B. 

End  Pins 

No 

Pins  in 
Line 

(Centi- 
meters) 

II 

Ill 

IV 

V 

Averages 

!i 

(25) 

IZ 

•g-s 

1 

5 

16 

61 

64 

Ph 

|§l 

1.5 

28 

27 

76 

41 

o 

2 
2.5 

58 
!K5 

8 
32 

60 
57 

30 
26 

a 

^2 

3 

99 

60 

52 

26 

g 
u 

Averages 

57.2 

28.6 

61.2 

37.2 

46.0 

Iz; 

ISi 

O 

(30) 

w  6    . 

1 

+  84.2 

+  43.5 

+  1.5 

-  8.7 

2; 

gi^'g 

1.5 

+  58.8 

+  24.8 

-9.0 

-14.9 

0 

0)    t.    fl 

CM  2  ^ 

2 
2.5 

+  30.5 
+   1.5 

+  25.1 

+  22.8 

-8.3 
-2.0 

-17.8 
-20.0 

O 

3 

+      .2 

+   4.3 

-8.0 

-22.0 

Averages 

+  35.0 

+  24.0 

-5.2 

-16.7 

+  9.275 

!i 

(35) 

u 

1  2  1 

1 
1.5 

30 

57 

65 
30 

78 
43 

90 
36 

,     65.7 
43.7 

!5 

'"  al 

2 

51 

47 

45 

40 

45.7 

2  ^  & 

2.5 

68 

49 

51 

48 

54.0 

0 

3 

82 

78 

60 

56 

69.0 

Averages 

57.6 

55.1 

55.4 

54.0 

55.62 

m 
H 

is 

(40) 

S 

1 

+  42.1 

+  16.4 

+  11.2 

+   7.0 

+  19.2 

g«^ 

OJ    t.    S 

Ph  2  -^ 

1.5 

2 

+  4.7 
+   1.0 

+   6.7 
+   1.1 

+    1.2 
-  9.3 

-  9.9 
-11.2 

+     .7 
-  4.6 

•-S 

2.5 

+   3.6 

-  6.8 

-  7.3 

—  12.3 

-  5.7 

o 

3 

-  3.2 

-  6.1 

-11.1 

-14.1 

-  8.6 

Averages 

+  9.6 

+  2.5 

-  2.5 

-  8.1 

+   2.0 

OUR    NOTIONS    OF    NUMBER    AND    SrACE. 


11 


Table  3.      Experiment  A.  —  Pins  in  straight  line. 
FOREARM. 


Sdbject 

Distance 
between 

Average  of  X.  P.  L.  and  B. 

End  Pins 

No 

.  PrNS  IN 

Line 

(Centi- 
meters) 

II 

III 

IV 

V 

Averages 

M 

(45) 

2 

0)     w 

1 

26 

31 

45 

23 

P^ 

■§  2  .s 

1.5 

39 

34 

37 

19 

o 

«  S  s 

2 

54 

37 

31 

20 

« 

2  "-  rt 

2.5 

59 

39 

41 

13 

ea 

P 

3 

64 

27 

30 

14 

Averages 

48.4 

33.6 

36.8 

17.8 

34.2 

'A 

O 

s 

(50) 

—   o     . 

1 

+  70.0 

+  18.1 

-11.7 

-27.3 

»S 

g£| 

1.5 

+47.0 

+  9.3 

-14.2 

—27.7 

S 

2 

+  43.0 

+   7.1 

-15.1 

-23.9 

o 

Q 

PM    g-^ 

2.5 

+  38.0 

+     .3 

-18.3 

—29.9 

3 

+  20.0 

+     .9 

-21.1 

-29.1 

Averages 

+  45.4 

+   7.2 

-16.1 

-27.6 

+  2.225 

8| 

(55) 

4>    ■" 

M  Q   —■ 

"3  1  » 

1 

20 

35 

42 

46 

35.7 

m 

1.5 

47 

33 

29 

27 

44.0 

S5 

H 

2 
2.5 

55 
41 

50 
41 

44 
39 

40 
40 

47.2 
40.2 

3 

57 

40 

49 

48 

48.5 

Averages 

44.0 

39.8 

40.4 

40.2 

43.16 

en 

H 

3 

(60) 

? 

■s"  '^  ^ 

1 

+  69.1 

+  56.0      +44.3 

+  40.0 

+  52.4 

0> 

g^^ 

1.5 

+  33.1 

+29.4      +25.1 

+  21.0 

+  27.1 

L3 

2 
2.5 

-  .2 

-  4.3 

+     .5      -  1.6 
-  6.9      -   7.3 

+      .9 
-  9.0 

-  .1 

-  6.9 

3 

-  9.1 

-12.3      -14.9 

-15.1 

-12.8 

Averages 

+  17.7 

+  13.3      +  9.1 

+   7.6 

+  11.9 

12 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


Table  4a.     Experiment  A.  —  Pins  in  straight  line. 
ABDOMEN. 


Subject 

Distance 
between 
End  Pins 

Averages  of 

N.  P.  L. 

AJST)  B. 

No.  Pins  in- 
line 

(Centi- 
meters) 

II 

III 

IV 

V 

VI 

Averages 

8 

03 

(65) 

ft 

^ 

O'-i 

1 

7 

32 

59 

55 

£.^ 

1.5 

8 

21 

57 

52 

oft 
o  ft 

2 

11 

16 

55 

47 

^  ^ 

2.5 

10 

20 

51 

42 

5 

3 

29 

28 

44 

44 

^ 

II 

3.5 

34 

29 

44 

37 

h 

m 

4 

52 

30 

49 

35 

o 

s 

4.5 

08 

36 

41 

26 

d 

5 

79 

38 

36 

28 

14 

a 

Averages 

34.1 

27.8 

48.4 

40.7 

14.0 

33.0 

^ 

> 

(u 

o 

H 

(70) 

6 

1 

+  109.0 

+  34.1 

+  12.1 

-13.3 

O 

1.5 

+  09.4 

+  30.8 

+  12.4 

-13.9 

1-5 

2 
2.5 

+  00.1 

+   78.8 

+  38.7 
+  37.2 

+  10.1 
+   9.1 

-14.9 
-16.2 

3 

+   62.3 

+  20.1 

+  9.1 

-J7-4 

o 

3.5 

+   53.0 

+  30.1 

+  6.3 

-18.2 

a 

4 

+  42.4 

+  25.1 

+  4.2 

-21.4 

4.5 

+  31.2 

+  11.4 

-  6.1 

-24.9 

Ph 

5 

+   21.2 

+      .2 

-12.1 

-25.5 

-22.2 

Averages 

+   6 

+  27.3 

+   5.2 

-18.4 

-22.2 

+  11.44 

OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


13 


Table  4b.     Experiment  A.       Pins  in  straight  line. 
ABDOMEN. 


Subject 

Distance 
between 

Averages  of 

N.  P.  L 

.  AND  B 

End  Pins 

No.  Pins  in 
Line 

(Centi- 
meters) 

II 

III 

IV 

\' 

VI 

Averages 

1 

(75). 

>^ 

a-A 

1 

42 

53 

66 

78 

59.7 

£l 

1.5 

25 

24 

31 

30 

27.5 

11 

2 

28 

20 

28 

24 

27.2 

■^l 

2.5 

28 

28 

26 

25 

26.7 

Soi 

3 

28 

27 

27 

25 

26.7 

a 

1^-5 

3.5 

23 

21 

18 

20 

20.5 

1 

4 

27 

25 

22 

23 

24.2 

< 

4.5 

2() 

26 

24 

20 

24.0 

tf. 

5 

48 

44 

40 

39 

38 

40.2 

'"' 

Averages 

30.6 

30.8 

31.3 

31.6 

38.0 

30.74 

C 

T. 

H 

ie 

» 

5/ 

1 

+  37.0 

+  29.8 

(80) 
+  21.0 

+  16.1 

+  26.0 

i-s 

o 

1.5 

+   8.0 

+   8.4 

+   3.3 

-  8.1 

+   3.1 

p  3> 

2 

+  12.8 

+   9.9 

+   1.0 

-    1.2 

+   5.6 

2.5 

+ 12.3 

+    1.1 

-  3.1 

-   6.2 

+    1.0 

iM   3 

3 

+  13.2 

+   9.2 

+      .5 

-     .3 

+   5.6 

C 

3.5 

+  12.0 

+    8.2 

+      .6 

-     .4 

+   5.1 

C 

4 

+   8.9 

+     .4 

-     .4 

-     .4 

+    2.1 

v 

4.5 

-     .8 

-     .5 

-  4.2 

-  7.7 

-  3.3 

I 

5 

-11.7 

-12.5 

-14.7 

—  15.5 

-16.2 

-14.1 

Averages 

+  10.3 

+  6.0 

+     .4 

-  2.6 

-16.2 

+  3.46 

14 


OUR    NOTIONS    0¥    NUMBER    AND    SPACE. 


Table  5.      Experiment  A.  —  Pins  in  straight  line, 
(a)  FOREARM.     Pressing  evenly,  three  times  only. 


SlIBJECT 

Distance 
between 

Average 

OF  N. 

AND  P. 

End  Pins 

No.  Pins  in 
Lute 

(Centi- 
meters) 

II 

III 

IV 

V 

Averages 

8  S 

(83) 

"=>  -s. 

Ph 

0)    *i 

^  2  £ 

1 
1.5 

13 
14 

21 

22 

35 

26 

33 
34 

o 

2 
2.5 

26 
20 

17 
22 

29 
34 

40 

32 

n 

3 

41 

21 

40 

32 

Averages 

22.8 

20.6 

32.8 

34.2 

27.6 

^ 

m 

o 

£ 

(86) 

H 

*^'  6    . 

1 

+  78.0 

+  26.0 

-  5.2 

-17.6 

Z; 

H 

^  c^  * 

1.5 

+  79.5 

+  26.7 

-     .8 

-16.4 

0 

(P    (-    s 

2 
2.5 

+  75.5 
+  71.5 

+  28.5 
+  21.2 

-  4.2 

-  4.1 

—  14.4 
-18.0 

o 

3 

+  66.5 

+  21.0 

-  4.5 

-16.7 

Averages 

+  74.2 

+  24.7 

-  3.8 

-16.6 

+  19.625 

8^ 

(89) 

?  2l 

1 

12 

6 

13 

15 

11.5 

u 

1.5 

28 

16 

16 

12 

18.0 

Iz 

r  Si 

2 

29 

25 

32 

25 

27.7 

H 

2.5 

42 

33 

25 

26 

31.5 

^ 

Ti 

o 

38 

41 

40 

40 

39.7 

b 

Averages 

29.8 

24.2 

25.2 

23.6 

25.86 

H 
>5 

o 

a 

(92) 

u 

«    w      . 

1 

+  86.5 

+  104.0 

+ 106.0 

+  98.5 

+  98.7 

1  £  S) 

1.5 

+29.5 

+  41.1 

+ 

29.8 

+  55.8 

+  38.8 

D 

t.  o  "2 

2 

+   1.6 

+     2.8 

+ 

6.5 

+  9.9 

+   5.2 

'-s 

P4    O    ^ 

2.5 

-  2.2 

-     2.7 

1.1 

-  2.5 

-  2.1 

w 

o 

3 

-13.4 

-   13.6 

— 

13.5 

-15.6 

-14.0 

Averages 

+  20.4 

+  26.3 

+ 

25.6 

+  19.2 

+  11.94 

OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


15 


Table  6.     Experiment  A.  —  Pins  in  straight  line. 

(b)   FOREARM. 

Attention  given  only  to  number  points  felt. 


Subject 

Distance 
between 

Average  of  N. 

AND  P. 

End  Pins 

No.  PlN.S  IX 

Line 

(Centi- 
meters) 

II 

Ill 

IV 

V 

Averages 

I  2 

(95) 

1?; 

1 

84 

21 

11 

0 

P4 

•3  o  j; 

1.5 

81 

25 

7 

1 

o 

iP 

2 
2.5 

79 
74 

27 
23 

14 

27 

0 
5 

i^ 

3 

79 

24 

27 

2 

S 

a 

Averages 

79.4 

24.0 

17.2 

1.4 

30.5 

'•^ 

»5 

in 
O 

s 

s 

(98) 

jJ    o      . 

1 

+  23.1 

+  12.4 

-27.0 

-43.6 

K 

gi^s 

1.5 

+  19.8 

+ 13.0 

-•27.2 

-48.9 

g 

^    o  -^ 

2 

+  14.4 

+  14.2 

-32.9 

-47.2 

Q 

(^  2-^ 

2.5 

+  19.5 

+  1G.9 

-33.5 

-47.0 

o 

3 

+  14.8 

+  14.7 

-26.6 

-53.1 

Averages 

+  18.3 

+  14.2 

-29.4 

-48.0 

-11.225 

«   "ai          "O' 

5 1     J 

(101) 

00    0<          A 

, 

i  0.2  & 

1 

84 

62 

55 

51 

of  ti 

3  ma 
mes 

1.5 

81 

54 

56 

58 

2 

79 

60 

60 

48 

4*  a  5 
.o   »    ?  o 

1  a     - 

2.5 
3 

74 
79 

63 
54 

42 
37 

63 
52 

^ 

.a,    a- 

Averages 

79.4 

58.6 

50.0 

52.4 

IG 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


Table  7.      Experiment  A.  —  Pins  in  straight  line, 
(c)  FOREARM.     Test  of  improvement  through  practice. 


Subject 

Distance 
between 

Average 

OF  N. 

AJfD   P. 

End  Pins 

No 

.  Pins  in 
Line 

(Centi- 
meters) 

]I 

111 

IV 

V 

Averages 

8  1 

(104) 

z 

£  '-5 

1 

12 

30 

50 

28 

Ph 

■c  S  ^ 

1.5 

26 

24 

56 

26 

o 

■«    11 

•) 

38 

34 

48 

12 

K 

a;    0.  g- 

2.5 

50 

28 

52 

12 

W 

1^ 

3 

64 

36 

48 

14 

p 

Averages 

38.0 

30.4 

50.8 

18.4 

34.4 

k( 

C 

■s 

(107) 

o  -o 

1 

+  84.0 

+  22.6 

- 

-  8.0 

-26.0 

1^1 

1.5 

+64.0 

+  18.0 

- 

-11.0 

-19.6 

S 

u     u     ^ 

2 

+  50.0 

+  21.3 

- 

-10.5 

-23.6 

C 

^  -•! 

2.5 

+40.0 

+  8.0 

- 

-  9.0 

-30.0 

a 

o 

3 

+  24.0 

+     .3 

- 

-  8.0 

-26.4 

Averages 

+  52.4 

+  14.0 

- 

-  9.3 

-25.1 

+  8.0 

(110) 

H 

"§  S  .2 

1 
1.5 

30 
32 

24 
30 

30 
34 

28 
36 

29.5 
33.0 

9^ 

'«  s  1 

2 

28 

50 

42 

38 

39.5 

H 

s  S^ 

2.5 

40 

32 

30 

46 

37.0 

R 

h 

1^ 

3 

84 

86 

74 

68 

78.0 

Averages 

44.0 

44.4 

42.0 

43.2 

43.4 

8 

(113) 

^  .23 

1 

+  48.0 

+  60.0 

+ 

55.0 

+  53.0 

+  54.0 

o 
a 

s  5  '^ 

1.5 

+  25.3 

+   20.6 

+ 

22.0 

+  21.3 

+  23.8 

2 

+  20.0 

+     9.5 

+ 

9.0 

+   7.0 

+  11.4 

•-S 

f^  o  •- 

2.5 

+  3.6 

-     2.0 

— 

4.8 

-  1.6 

-  1.2 

U 

w 

o 

3 

-  2.6 

-     2.3 

— 

11.3 

-  5.6 

-  5.4 

Averages 

+  18.9 

+   19.2 

+ 

12.2 

+  14.8 

+  16.5 

OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


17 


Tables  8  and  9.     Experiment  A.  —  Supplement. 


STRAIGHT-EDGE. 


OP  TBrR^* 


FOREARM. 


(«) 

(0) 

(d)  Regular. 

(e)  Pressing  evenly 
three  times  only. 

Distance 

Average 

Average 

OR  Length  of 

OF                   1 

OF 

Straight-Edge 

N.  P.  L.  AND  B. 

N.  P.  L.  AND  B. 

8  s 

(1 

4) 

(11(3) 

-o  .5 

1 

53.0 

48.7 

H 

1.5 

50.0 

39.7 

f. 

T  S    0. 

2 

60.0 

46.5 

H 

IS" 

2.5 

72.0 

44.5 

O 

H 

3 

83.0 

55.5 

Averages 

63.8 

47.0 

(1 

I.-.) 

(117) 

s 

*i  .2 

1 

+  31.4 

+  34.8 

§-  S) 

1.5 

+   8.9 

+  10.6 

3 

"    ®  -3 

2 

+  2.7 

+    1.3 

(^2-- 

2.5 

-     .6 

-      .8 

3 

-  7.8 

-    1.5 

Averages 

+  6.9 

+    8.9 

18 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


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i-i  in  fn -^  ^  fn  fn 

IJH  P^ 

1 

H  Ph  P^"<  Pt| 

f^f^ 

PmP^ 

0 

•pailddn  soim)  fK1I 

jod 

j       -jojja  JO 

IIIM  J 

M 

EXPEEIMEXT    B. 

WITH   pixs  SET  IX  triax(;l?:s  axd  squares. 

Apparatus.  —  Triangles  and  squares  of  proper  dimen- 
sions were  cut  from  specially  heavy  "trunk"  cardboard. 
The  pins  were  thrust  through  the  board  at  right  angles 
to  its  surface,  care  being  taken  to  have  their  points  lie 
perfectly  in  the  same  plane. 

To  the  "  distance  "  and  "  number "  categories  of 
Experiment  A  was  now  added  the  category  of  "  figure." 
The  two  "  figure  "  categories  used  in  the  present  experi- 
ment were  those  of  the  triangle  and  the  square. 

The  same  "  distance  "  categories  were  used  as  before  in 
A;  and,  as  before,  these  measured  the  distances  between 
the  end  pins,  or,  as  it  Avould  be  in  this  case,  measured 
the  outside  lines  of  the  triangles  and  squares. 

The  ''number"  categories  here  used  will  easily  be 
understood,  while  referring  to  the  horizontal  headings 
of  the  tables,  if  I  explain  that  "HIT"  means  a  triangle 
with  a  pin  in  each  corner  ;  "  IV  T,"  a  triangle  with  a 
pin  in  each  corner  and  one  in  the  center  of  the  triangle  ; 
u  yi  i'^'"  a,  triangle  with  a  pin  in  each  corner  and  one 
bisecting  each  of  the  three  sides ;  "  VII  T,"  a  triangle 
with  six  pins  arranged  as  above  and  still  another  pin  in 


22  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

the  center.  "IV  S,"  ''VS,"  ''VIIIS,"  and  "IX  S  " 
indicate  squares  similarly  arranged  to  the  triangles. 

Method.  —  This  differed  from  that  of  Experiment  A 
only  in  that  the  pins  were  now  applied  to  the  subject 
by  some  one  other  than  himself,  he  not  being  able  to 
handle  this  apparatus  without  learning  thereby  some- 
what of  the  size  of  the  board  which  held  the  pins. 

The  subject  now  had  three  judgments  to  make 
for  every  application.  He  usually  made,  and  always 
announced  these,  in  the  same  order,  and  as  follows,  I.e., 
"distance,"  "number  of  pins,"  "figure." 

E.rplanation  of  the  Tables.  —  These  tables  are  much 
like  those  of  Experiment  A,  except  that  a  fifth  main 
horizontal  division  has  been  added,  giving  the  number 
of  correct  judgments  as  to  the  figure  in  which  the  pins 
were  arranged,  calculated  from  one  hundred  applications 
to  each  person. 

Also  for  the  better  comparison  of  the  result^  from 
the  "  triangles  "  with  those  from  the  "  squares,"  a  more 
complicated  arrangement  of  "  averages  "  in  the  several 
minor  or  sub-tables  was  requisite.  This,  however,  will 
be  clear  if,  referring  to  Table  11,  I  explain  that  any 
figures  found  in  the  vertical  column  marked  "T."  are 
averages  of  the  foregoing  figures,  in  the  same  horizontal 
line,  to  be  found  under  the  four  vertical  columns 
marked  "Triangles";  and  those  under  "' S.,"  similarly, 
are  averages  for  the  foregoing  figures  under  "  Squares." 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.  23 

Tables  11,  12,  13,  and  14  are  "  regular "'  in  method 
and  compare,  respectively,  with  Tables  1,  2,  3,  and  4  of 
Experiment  A. 

Table  15  is  "irregular,"'  in  that  the  pins  were  per- 
mitted to  be  pressed  only  three  times  in  succession,  as 
in  Table  5. 

Table  16.  Attention  concentrated  solely  on  number 
of  points  actually  felt,  as  in  Table  6. 


24 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


o 

0 
<! 

O 

o 

Q 
'A 
< 

H 
OJ 

H 

4) 
< 

< 

CO 
OS 

CO 

0 

r 

CO 
CO 

Ci 

01 

0 

f 

ID 

<3 

00 
Oi 

0 

r 

•a 

Ci  05 

o 

CO 
Oi 

CO    T-H 

1  1 

0^ 

1 

n 

'B 

>> 

00  05  c;  CO  c; 
ci  ci  C5  c;  c. 

CO 

CO   T-H   1-H  rH   i-H 

rH 

1        1        1     +      1 

1 

>- 

O  CO  C5  o  o 
^^    Ci  Oi  Oi  o  o 

CO                             rH  rH 

ci 

(N  -*<  (M 

+ 

(0 
10 

Fi 

r' 

cc  c;  C'  o  c: 

O  C5  C'  o  o 

c: 

"^^    CO  ^1 

+ 

pi 

0 

h> 

t^  00 

05  C» 

Oi 

1    1 

1 

1 

K" 

CO  I^  CO  05  Ol 

cs  05  oj  05  c; 

CO 

07    0    0?    T-H    ,-H 

Ol 

M    1    1  + 

1 

1 

>; 

O  CL  o  o  o 
Ci  C:  c:  O  C 

Ci 

C- J  CM  T-H 

+  +  +  "" 

+ 

r-i 

H 

1 

1— 1 

CO  c;  05  o  o 
Oi  o:  oi  o  o 

T-l  rH 

oi 
Oi 

CD  CC  01 

01 

+ 

c 

T-l  rH  (M  (>i  CO 

45 

rH  r-i  5^1  oi  CO 

u 
a) 
> 
< 

O 
an 

Ph 

CM 

d 

!2i 

saunj  001  jad  ^ijoaj 
-joo  paSpnC  saun^  -o^ 

•paSpiiC 

sat  J  'o^  JO  jcua  JO 

•;«80  jaj; 

•SNij  >iO  Haa 

MOJ 

^  ao  SiNHKOanp 

OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


25 


(N  p  'TJ  O  ■* 

00  t-^  t-^  ci  oi 
C5  ci  oi  C5  c; 


0*  c^  t~  ^1  "-O    Ct 

oc  t-^  :d  ci  c^   00 
c:  ~.  C-.  a  c.  a 


CO  ^  1-^  CO  ci 
C;  02  Oi  O  <35 

« 

i-H  rt  (M  !N  ' 


sami)  001  Jdd  X[)33J 
-joo  paSpnt  sanii)  •o^j; 


o 

i-H      cc      o 
+  "  +  "l" 

o 

o  c;  o  -^  o 

TTT+T 


£S  +  +  + 


O  CC  o 

^  ^  ^  o  o 

+  +  -F 


uO  p  lO  « 


+ 


+  +  + 


CO  CO  ^1 

+  +  + 


1— I    r-<    3<)    l^i   CO 


+ 


•poSpnC 
aouB^stci  JO  .lojaa  jo 


•aoxvxsiQ  JO  sxxanoa.if 


O     r3 
be    0} 


«     S 


^      > 


W      s 


•panddB 

sauni  001  -lad  sjuaiu 

-SpnC  ^oajjoo  jo  'oji 


■aH.iyij 
JO   SXXaKuu.if 


26 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


o 

"ej) 

S 

■pj 

4J 

.a 

+j 

P 

m 

f^, 

N 

0) 

•S 

1 

o 

n 

h 

*i 

a 

o 

o 

a 

«5 


^5  ""J 


Tt*  (M  00  O  O  -^ 
,  lO  >0  ?0  l^  CO  CO 


I— i  ,— I  5^1  (N  CC  CC 


sarai?  001  J8d  ^noeJ 
-.loo  pagpnC  sauni  'on 


+ 


+ 


+ 


02  c<!  T-<  o 


C.  C:  i-H  O  lO  i-H 
CO  Ci  O  rH 

I  I T+++ 


C:  CO  T-H  o  CO  05 

00  C^  C:  CO 

CI 

+++++ I 


■  lO  ^^  r—  r-  O  O 


++++++ 


^  ■*  ,-1 


I  I 


+ 


+ 


O  t~  5<l  C5 


+++ 


O  rH  05  00  CC 

>0  i-H  O  CO 
CO  --I  i-l 

+++  +  + 


I  00  O  CO  (M 


++++++ 


t-H  CM  (M  CO  CO 


+ 


+ 


+ 


•paSpnC 
suid  "OK  JO  JOjjg  JO 

■!)U80  jaj 


•SNijj  JO  aaaKa^  slo  sxKarcoanp 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


27 


(N  c<;  00  -*  o  CO  t-; 
o  ci  i-H  od  CO  CO   00 
CO  t^  t~  CO  t~  Ci  t^ 

t—  'M  LO  -N  -^  CC 

-*  .-H  c^i 

+   1     1   +   1     1 

1 
+ 

<J<j  !>;  CO  p  00  p 

Ci  CD  CO  ci  C^  C5 
00  Ci  0  Cl  Ci  C: 

CD 

a> 

!>;  CO  CO  p  p  <N 

CO  CO  ^  o  o  1:0 

CO  t^  t^  t^  t^  Oi 

CO 
ci 

0  tM  CO  ■*  0  I- 
CO       '  '>!       ■       '       ' 
+     1       1     +     1       1 

p  p  I-  p  p  p 

T-H  CO  Ci  d  d  d 
CI  CI  CI  0  0  0 

CO 
CO 

0 

t^  0  CO  t-;  0  (M 
00  0  --H  CO  --!  ^ 
CO  CO  t^  CO  I--  CJ 

CD 

CO  ->]  t-  i-H  10  p 
0^  'N  (>i              '  rH 
+     1       1     +     1       1 

1 

CO  CO  0  t^  t:^  t^ 

t-^  LO  t-^  ci  d  d 

CO  c;  CI  CI  C:  c: 

p 

0 

CO  uo  0 
CO  t^  CI 

d 

CO 

+  1.8 
-  .0 

+ 

1— 1  1—1  rH 

0 

0 
1—1 

0  CO  ^  CO  0  CO 
Cw  t^  I^  CO  lr~  CI 

ci 

0  CO  UO  CO  0  CO 

+  1    1    M    1 

+ 

t^  CO  cs  0  0  0 
CO  ci  ci  0  0  0 

1— 1  i-(  r-< 

CO 

c;  0  •>!  0  I— 1  00 

UO 

->!  CO  0  0  0  0 

,^  0  c;  0  0  0  0 

CO                      ,-(  r-H   ,-1  ,-H 

CO 

CO 
CO 

CO  0>  01  CO  ■*  CO 
.-^  00  t:~  CO  CO  t^  Ci 
CO 

«  +  1  T  1  1  1 

Cr5 

+ 

Z2- 

■ — ■  0  CO  ^  'CO  '30  C:    l~ 

^  30  0  0  0  0 
000000 

00 

06 
0 

CO  CO  t--  CO  CO  1^ 
t-  r-  i^  t^  i^  Ci 

CO        CO 

+  +    1    +    1      1 

+ 

t^  (M  rH 
CO  t-C5 

CD 

CO  3D  LO 

I-H         '    T-H 
1          1          1 

7 

000 

0  0  O' 

t-; 

1 

CO  t--  0  TJH  CO  CO 

r- 

r-f'*  r-  0  0  0 

CD  0  0  0  0  0 

0 

^  CI  ?N  CO  0  0 

0  I-  «o  CO  I-  c: 

CO  --1 

+ 1  T++ 1 

1 

01 

r^  -rr   i-H    'M    X   r-H 

^ 

1-  0  t^  0  0  0 

CO  C5  Ci  0  0  0 

1—1  1—1 

LO 

lO 

0 

CO  0  0  >-0  ^  CO 

cooot-  CO  t~  CO 

+  1  1  +  1  T 

+ 

t-  r-H  -M  c:  ^  CO 
CO  CO  00  CO  t-  C-. 

p 

CO 

TtH  CO  rH  p  Tt;  p 
"ti        ■  CO  T-^ 

++++ 1 1 

LO 

+ 

■^  t~  I—  0  0  0 

ci  ci  ci  0  00 
I-l       1—1 

00 

0 

10         0         lO 

1-1  r-!  cq  c^  CO  CO 

< 

0           LO           W5 

1-H  1-H  Ol  C^  CO  CO 

ID 

LO            LO            LO 

1— 1  i-<  "M  0<i  CO  CO 

<1 

saniij  001  Js*!  s}naiu 
-SpnC   loajjoa  jo  ■oj^ 

■paSjinC 

90in!js!(T  JO  jojj^,i  JO 

•^1190  jaj 

•p8;[ddij 

saniii  001  Jad  Sineni 

-SpnC  ?03JJOD  j»  -0^ 

•30MVXSI( 

I  -J 

0  sxxaivoanjp 

.io  sxxaKoaax' 

28 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


O 

<! 

O 

2 

-i  he 

CO 

+ 

0) 
C3 

> 

o 

CO 

CO 

+ 

s 

CO 

CO 

+ 

3 

K 
to 

E^ 

r-l  O  .-H 
CO  CO  lO 

CO 

d 

CO 

^  O  r-; 

00   ^   i-H 

TTT 

CO 
1— I 

1 

^  Oi  05  t~  CO  t^ 
-#  -^  CD  lO  ■*  i-i 

CO 

o  o  CO  T-i  ci  c^ 
,-H  cq  ■-<  --H    ■  o4 

iM  iM  (M 

1    1    1    1    M 

T 

K' 

>0  r-i  05  CO  -r*<  O 
^^  CO  -^  -"i*  »o  o  t^ 

CO  O  CO  r-l  CO  (01 

'^'  d  d  <6  c^^  n 

Oi 
00 

+ 

4J 

n 

1 
1 

>; 

-t*  'M  O  CO  O  rH 
CO  i-O  -^  CO  lO  CO 

0>] 

^  '^  ^  q  o  CO  q 
oi  00  00  r-^  d  i-^ 

CO  (M  Ol  Ol  1— ' 
++++++ 

Ci 
CO 
01 

+ 

2 

P 

o  >o  .-H 
CN  <M  CO 

CO  q  q 
00  d  d 

T-H  rH  (N 
1        1        1 

CD 

d 

1 

1— 1 

-^♦<  CO  O  CO  O  O 
CO  CO  LO  lO  lO  -^ 

CO 

o  .-H  o  o  o  o 

d  00  i-O  CO  T)i  -tlH 

1 TTT  u 1 

CO 

d 

T 

1 

\-- 

CO  >0  lO  O  i-H  CO 
(M  CO  ■*  lO  lO  ■* 

oi 

■*  (N  (01  rN  o  ■* 

■-;  rH  d  CO  00  00 

Ol  1-1 

++++++ 

00 

d 

+ 

TO 
H 

5 

CO  -*  O  O  CO  (Tl 
(N  CO  ■*  Ttl  Tti  O 

o 
d 

c^_  <z> '^_  o^_  ^_  c:^^ 
o6  lO  00  d  d  CO 
lo  ■*  (oq  (M  (01 1-< 

++++++ 

CO 

+ 

1 

f 

--a 

lO          »C          »C 
T-H  .-H  (M  S<i  CO  CO 

SB 

uO         lO         lO 

rH  ^  (M  Oi  CO  CO 

-1^ 

o 

Oh 

2 

d 

sauin  001  -laci  X^oaj 
-.103  paSpiif  sauii)  •ojs[ 

•paSpuC 
siiij  -on:  jo  jojja  JO 

•SxMij  >io  Haa 

KllJ 

sj  ^o  sxNaKoanp 

OUR    NOTIONS    OF    NUMBER    AND    SPACE.  29 


O  CO  O  'M  CO  'O 


t-  cr  •TO  q  o  o 

lO  t:;  r-  ■— '  1^  o 
;~  t^  1-  t—  t^  c: 


1^  CO  t~  >1  lO  O 

CO  oi  o  t^  oi  CO 
J—  I—  t^  o  t^  c; 


o      »o>      o 
1-H  i-J  0^  oi  CO  CO 


•pajlddtj 

sauin  001  ■18'J  ^1%03J 

-joo  paSpnC  sauiij  'Ox; 


lS>  O  lO  00  ' 


+  +  +  +  I 


+ 


CO  c;  X  q  r-;  i^ 

OC   r-^  rH  (>j 

+++++ 1 

CO 

o4 

+ 

b-;  q  01  q  CO  I— _ 

Oi       ■  r-I  i-H 

+  +  +  +  1     1 

q 
+ 

+3.1 
—1.8 
-2.0 

1 

o  00  q  rH  (?^ 
uo  1-i  (>i    '    ' 

CO 

++  +  +  +    \  + 


o  o 

O  C  --^  t- 

o 

C-.  ^  O?  CO  .-1 

++++ 1 1 

CO 

+ 

C  CO 

-*  X  c;  (M 

Ol 

+++++ 1 

+ 

3<l  CO  0<1 

cc 

+     1       1 

+ 

•rf(  ■-.  O  O  --I  "* 

>o 

CO  O^   04  .-1  i-H 

+     1       1    +     1       1 

+ 

,-;  oi  X  q  CO  q 
+++++ 1 

oi 

+ 

1-  o  Ci  't :;:  oi 

:^ 

CO 

+++++ 1 

OI 

+ 

O  O  lO 

^  ^  ^  (?i  CO  00 


-i; 


•paSpwC 

saotre^sjci  jo  joajg;  jo 

■%a9o  jaj: 


•aoxvxsi(i  JO  sx.N;aKoa;ip 


O  tN  O  Oi  (N  O 


o  CO  CO  q  0^1  o 

CO  :o  lo  00  ci  00 


O  O  t~  t-;  01  lO 

00  CO  CI  Ci  c:  O 


O         L.O         o 

1 1— H  o^  oi  CO  CO 


•paiiddu 

S9uir}  001  -lad  jS(103J 

-JOO  pa3pnC  sauiii  -o*! 


•aH.ioij 
ao  sx^aKOU-ix" 


30 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


1 

o 

Q 

m 
< 

"A 

O 

0 

i 

< 

CO 

ci 

+ 

41 

<5 

© 

5i 

CO 
CO" 

+ 

2 

ni 

3 

C5 

+ 

'^ 

g^^^^g 

CO  CO  t-  00  CD  in 

•^  lO  t-^  00  Ci  00 

TT 1 1 1 1 

© 
1 

•n 
a 

1^ 

coo."]i-icoTt<eo(M>oco 

© 

CO 

O  IT]  ©  t-;  C^l  C^  C^l 

ocoo6i-^si'*»io©© 

TTT 1 1 1 1 

© 
+ 

^ 

CO 

(N  tC  C-]  01  ©  ©  ©  00  t-; 

r-i  rH  r-I  LO  -t  C-i  't  -*  oi 
?]  i-H  rH  C-1 

ml     +     1     +    +    +    +    +    + 

1 

n 

■4-) 

a 

0) 

!"• 

i 

"""ic  in  »q  ©  ©  lO  ©  lO  iq 

Oi'l^in-OCO-Tt^OOCOrji 
r-(T-li-ICO'*<MCO-^ 

+++++++++ 

© 
+ 

■ 
o 

r' 

00  »  1-1  3~.  (M  r-l 
r-i  C-J  T-I(M  CO 

© 

r)J  tH  >*  CO  t);  CO 

in  in  i-^  ©  ^  ci 

Cl  O-I  i-H  C'J  CI  C-l 

1    1    1    1    1    1 

in 

ci 

1 

r' 

^  CI  •>*  CO  c~j  iM  w  e-1  00 

C0t-;COt-;CO©i-;TH© 

ci  -^  ci  rH  co"  "*  ci  co'  <* 

TTTTT  i^TTT 

00 

1 

(MiHOtOCOOOrtiT+lcc 

© 

©inioin©in©in© 
CO  c-i  CO  ci  ci  -o  r-i  i-J  in 

T-H  cq  rH  rH 

1   +   1    +  +  4-  +  +  + 

CO 
+ 

H 
H 

S 

CO 

>* 

©  »  •^_  X  ©  CO  ©  CO  CO 

oi  'I'  -)<  in  -^  i-H  c5  in  co' 
CO  CI  01  -t<  in  ;o  CI  oi  i-H 
+++++++++ 

CO 

CO 

+ 

Ed 

>o      lo      ira      »o 

iH  rH  C-1  CJ  CO  ^0  •*  Tli  lO 

a) 

in      in      in      in 

rH  r-!  C^  Ci  CO  co'  ■*  •<*  in 

< 

o 

5r, 

c 

c 

sauii;  001  jad  Xnoaj 
-.loo  pgSpnf  sauu;  'Ofj 

•paSpnC 

siud  'oa  JO  jo.ua  JO 

•?u8a  aaj 

•SMij  io  aaai\ 

.a^ 

;  do  sxKaKOuiif 

OUR    NOTIONS    OF   NUilBER    AND    SPACE. 


31 


1-  L-  X  x  k-  —  ri  o  m  z: 

X  -^  r!  d  d  '^  ^  •*  <£  H 
■■c  -.c  --^  -^z  »-•;  lO'-c  ~  X  --c 

~    X    X    -^    t~   -f    Lt     ~;    — ; 

+        1          1        +       1       +       1+1 

1 

X  q  L-;  X 

i"^  X  ■#.  ?:  11 S 11 

"1 

o  q  t-;  L-;  L-;  ?i  c^]  iq  ti 

— <  •.£  «*  ci  IS  c;  id  •^  »r; 

r-.  r.  :-  -r  q  -^  '^.  •*  >>: 

t-^  -«•      '  C-i  IT  i-i         r-1  ^ 
+     1      1     +     1     +    +   +     1 

C-l 

+ 

yt£sg|g8g 

X 

r^_  q  c  c  rt  ir;  c  o  t-; 
d  t-^  id  1—  t-^  -r  t-^  cc'  id 
t-  13  -^  -.c  L";  I--;  o  :C  t- 

^ 
ri 

ct-;r:xqaqq'*q 

c-i  tt  rd    ■  c4  Ti'  r-i    '  e-i 

+  T  1  +  1  +  1  +  1 

q 
1 

•I  ^.  "^  ^. 

L* 

«3KEHg 

g 

rf  CC  X  CI  .-;  c^ 
c-i  rt  ?i  c-i  cd  c^ 

+   1   +  +  +   1 

+ 

gggggg 

g 

£5SS":h:;SS:3 

X 

x 

+  17.0 

-10.0 

+   2.0 

0 

+   7.4 

+  •  7 
-  2.2 

+ 

gg?:SS8||| 

3 

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X 

Or5iC?OCO-*(MT-IC<5 

S-+  1    1  +  1    1    1  +  1 

+ 

gtrSggggSg 

q 
id 

i2S^^SS5E=? 

- 

^^q  ^:  o  q  q  •*  o  •*  q 
-*  .-;  ?i  tt  rd  1-4    ■    ■  r-! 

+   1    1   +   1   +  +  +   1 

+ 

g?.£S|||S| 

i 

SSSis^S 

X 

CI  ?^  t-  X  O  C*l 

7-;  1-;  r-j  ■  c-i  cd 
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q 
T 

x  s  5  c  c  s 

t-; 

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S3 

q  cc  q  T);  q  X  c^  c?  q 
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+  T 1  +  1  +  +  +  1 

1 

--■^•lllils 

i 

i!Sd3;fS^l5I=t3 

X 

4 

q  -^  o  q  q  o  t-;  <N  oc 
c;  c  id  ri  I-!  i-<  i-i  ci  ci 

+  T  1  +  1  +  1  +  1 

1 

^g?.|lllll 

g 

SsSg^SggS 

s 

cd  X  ^f  o  rii  .^  .^  cJ  ci 

+  T  i     1  +  1  1  1 

q 
1 

2-^111111 

t-; 

lO       lO       «       o 
1-1  r^  ?i  ri  cc  rd  Ti-  rr  i.^ 

> 

< 

lO        O        lO        o 

T-l  .-;  -M   f  i  rO  cd   "*   Tj^  L- 

u 

V 

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1-        o        o        1.-5 

1-  r^  M  ~i  r:  ^:  ■*  •*  13 

m 

1 
u 

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sauiii  001  Jad  ^naaJ 
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■pagpnC 

OOUB^SjO   JO   JOJJ^   JO 

•p3![ddB 
-juo  paSpnC  saiuii  -ojj 

•aoKvxsi 

a 

JO  sxNaKoaaf 

io  sxiiSKoaap 

32 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


K 
<1 
H 
K 
O 
Em 

>> 

•g 
3 

» 
o 

S 

0) 

+-> 
s 

> 

a 

CR 
(U 

IH 

Oh 

1^' 
W 

3 

a 

11 

I* 

<1 

o5 

OJ 

T 

a; 

CB 

u 

to 
> 

< 

o 
S 

T 

01 

01 

+ 

02 

y. 

05   r-i 

q 

00 

ooo 
CO  t-; 

1  1 

1 

!0 

!> 

- 1-  CO  c:  t-  o 
■-1  'N  Ol  .-^  oo 

o 
(>1 

q  q  q  o  oo 
ci  c<i  -^  id  co' 

00  oo  CO  CO  oo 

1  1  1  1  1 

CO 

1 

> 

c;  t~  lo  c.  fM 

, — .          1— 1  (M         ■<*( 

c; 
o 

o 

0-1 

O  (N  Tj<  00  1— 1 
I-^  CO  O  -*■  CO' 

2  T  1   1   M 

q 
1 

(U 

If] 
ai 

> 

5i 

t^  >0  t^  00  o 

^  O-J  CO  -*  Tfl 

'CO 

^  q  O.  o  o  o 
c  c;  >d  c;  o 

+  +  +  +  + 

q 
+ 

1. 
1 

H; 

CO  o 

q 

00 

q  <N 

1  T 

1 

3 

- 

Of  lO  T-(  lO  o 
Of  (M  C^  Of  Tji 

00 

O  t-  t^  O   I-H 

^  ?j  ~"  ^  J^ 

1    1    1    Kl 

d 
1 

u 

3 

r- 

>-l   T-H   00   O   Ci 

-*  •*  <N  ?-1  CO 

00 

q  <z:_  o_  <z-_  <^^ 
+  +  +  +  + 

q 
+ 

> 

S 

cr  o  r~  00  o 
OO  CO  1— 1  oo  -* 

X 

00  O  O  00  0-1 

t^  r^  -^  CO"  oo' 
00  ■*!  Tti  O  C^I 

+  +  +  +  + 

d 
+ 

3 

T-i  --;  <M  CN  00 

Ml 

u 
> 

T-<  i-H  0>1  <N  CO 

o 

a-j 

PS 

Ph 

W 
K 

O 

2 
1 

saiiii^  001  J8d  :Ciioaj 
-joo  paSpiif  saun;  -ox 

"paSpuC 
S"!d;  'OM  JO  jo.i.ia  JO 

•sxij  Ao  Haai 

«n^ 

do  sxNareoanf 

OUR    NOTIONS    OF    NUIVIBER    AND    SPACE. 


33 


Cvj  O  t-;  50  O 

cc  oi  i-H  CO  CO 
■^  <N  ■^  CO  "^ 


r~  !^•3  O  i^  O 


rH  1-H  '^^  ^1   CC 


ssiui?  OOT  J8d  Xnoaj 
-joa  paSpnC  sauii}  'ox 


CO  CO  uO  CO  o 

CO 

^  CO  o  «  o 

Ol 

O                 ,-1  71 

+  +     1       1       1 

+ 

l~  lO  t-  »  c 

o 

Tjl  00  CO  ■*  CO 

•^ 

++  1  1  1 

+ 

OCiCO  O  i-H 

0^ 

CO          'if  O5C0 

Ttl                    I-H  (?q 

+ 1  1  1  1 

+ 

«D  CO 

o 

I^  -* 

•o 

I— 1  I— ■ 

1  1 

1 

O  O  O  -*  uO 

00 

to  iM  lO  ^  c: 

t-- 

O  1—           .-H  rH 

++ 1  1  1 

+ 

O  I—  O   (>1   r-H 

o 

o  o  o  o  GO 

>-o 

++TTT 

+ 

O  t^  O  ffl  ^ 

cc 

00  !M  — 1  r-l  O 

CO            .-H  r-l  <N 

++  1  1  1 

1 

O  CO 

■* 

O  .-1 

X- 

(M  C- 

1  1 

Ol 

1 

O  t^  00  Ol  --1 

'^ 

CO  O  r-l  CO  O 

O  i-i         (>»  CO 

+  1  1  1  1 

+ 

oco  o  -*  --1 

-* 

■*  lO  O  -^  GO 

Ol 

■*                  i-H  rH 

++  1  1  1 

+ 

o  o  o  CO  o 

o 

Ol  X  "w  -M  CO 

f—t 

CO                  1—*  T— < 

+  +  1   M 

+ 

1 1-i  oi  oi  ■ 


■pa2pnC 
8DUB}S!a  JO  Jojjg;  jo 


•aaxvxsiQ  jo  siKHivoa.if 


t^  O  O  01  Ci 

x> 

lO  iQ  lO  lO  lO 

o 

CO 

01  01  0) 

-o 

J-.  o  ^  cr.  CO 
CO  "^  "^  "^  o 

-*l 

CO 

CO 

CO  01  o 

^ 

--0  r-l 

O  C2  Ol 
iC  UO  <» 

03 

0 

s 

T-H  1-H  Ol  Ol  CO 


•pailddB 

saiun  (X)l  Jad  Xnoai 
-Joo  pd3[nif  soiui;  x 


34  OUR   NOTIONS    OF    NU]MBEE    AND    SPACE. 


<! 
H 
K 
O 

j) 
n 

3 

o 
)i 

V 

fl 
O 

0 

s 

1 

60 
§ 

1 

•4J 

b 
O 

35 

O 

« 
> 

s 
<; 

0) 

q 

+ 

a; 

s 

o 
id 

+ 

id 

S 

CO 

00 

1 

a" 

1 

^ 

oo 

o 

q  r- 

+  1 

+  ' 

® 
s 

B 

> 

o  o  o  o  o 

o 

q  q  q  Gc  o 

ri  CO  :n     '  i-h' 
t—  •N  i-H          CI 

+  +  +     1       1 

o 

00 

+ 

> 

_  ooooS 

q 

q  o  o  00  or  c<i 
cc  ^6  --6    "  o   -ri 

^^    ■^  CO                1-13^ 
2     +  +  +  +     1      + 

■4-> 
<U 

tn 

CD 

^ 

•go  O  rH  00  O 

CO 

^  qccq^t^TK 
c£  o  c<i  o  o<i 

t-  CN         .-.  T-i 

+  +  1  +  1 

+ 

1 
PQ 

> 

oo 

^ 

o  o 
o  d 

CN  CC 
1      1 

o 

00 

1 

1 

> 

o  o  o  o  o 

o 

q  q  q  rN  q 
-^  o'  ao'  >o  .-! 
■^  CO       i-i  cq 

+  +   1     1     t 

d 

+ 

J 

>> 

O  O  O  00  -M 

q 

(Z>^  <^_  <D  •^_  ^_ 

^  -^  ^  d  t-^ 

++ 1  TT 

0^ 

+ 

- 

OTtH  O  O  O 

q  i>;  q  00  q 

d  d  -^  00  oi 

lO         ^         i-i 

+  +  1    1    1 

CD 

+ 

O 

OS 

H 

a 

O 

Q  W 

i-H  i-i  oq  (>i  Of 

0^ 

> 

--1  i-i  tM  c<i  CO 

1) 

1^ 
6 

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■paSpnC 

SU'tl  'OiJ  JO  JOJJa   JO 

•SKij  jio  Haa 

Kn\ 

[  JO  sxNareouap 

36 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


(0 

(i5 
> 

o 

•S 
«r 

H 

o 

■4-> 

H 
H 

of 

3 

+j 

ta 

a 
o 

W) 
(D 

o 

a 

0 

is 
(« 

0 
■u 

§ 
XI 

01 
(U 
00 

nS 
o 

\ 
to 

in 

o 

H 

?;   . 

oo 

gQ 
i^  ^ 

0^ 

'^-'  ^J 

<^    • 

^6 
H  tc 

oi 

(»  & 

2« 

W 

« 
W 
(> 

C 

0 

O  !- 

t-;  CO  p  'O  iq  p 

CO  O  ->!  r-I  O  Oi 
CI  O  TjH  -*   fN 

I-        >0  CO  o 
CO  C;  -*l  tN  CN  i-H 

■  uo  CO    ■  d  t-^ 

1 +++ 1 + 

+ 

CO  o  o  p  ic  p 

CO  d  o  o  o  T-H 

C-.  1-  -*  -H  iM  --• 

d 

C^  p  CO  CO  CI  p 

'  d  r-i  CO  d  >.-f 

1  +++T  + 

+ 

ai 
t 

^  ^5  Ol  O  lO  p 

00  --!  i-^  CO  CD  ao 

Ci  lO  CO  CO  (>» 

CO 

>0  p  p  CO  CO  CO 

lO  lO  1-5  d 
1  +  +  +  +   1 

1-; 

CO 

+ 

H 

K 
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c 

y, 

O  O  CO  1-;  p 

CO  .-I  ci  d  CO  o 

—.  l^  CO  "*  (M 

CO 
CO 

S<j  O  uO  r-;  p  0-] 

'  i-H  CO  d  i^ 

1  1  1  1  1  + 

1-J 
o<i 

1 

(0 

a 

(Tl 

y- 

p  o  o  p  p 
CO  00  CO  CO  d  o 

C:  t tl  O  (M 

CO 

d 

r-H  lO  UO   r-<  t--  p 

"   -^  I-H  d   -^  CO 

r-H   ,-1  CO   ^ 

1    1    1    M  + 

1—1 
t-5 

1 

.s 

r- 

o 

i:^    <M  O  C^I  CO  ^  p 

i^  d  CD  ci  d  d  d 

Ci  CO  -*  '^  S^  i-l 

CO 

^   I-;  CO  p  >-;  p  cq 
(>j      '  d  00  d  i>^  <>i 

^++++ 1 + 

1-^ 

+ 

r- 

h^ 

^  lO  (M  I— 1  -"Jl  p 

d  oi  d  d  d  d 

O  CO  -*  lO  CO  CO 

T-H    O]   p   P   P   O 

"  d  CO  d  (>i  d 

Cl  CM  0-1  --I  i-H 

++++++ 

P 
I>5 

+ 

1 

K 
►J 

w 

!> 

o  r~;  CO  p  p 
r-^  d  lO  d  CO  o 
Ci  ^  CN  CN 

CO 
CO 

CO  CO  CO  lO  -^  o 
■  d  d  ci  d  00 

O)         ^ 

1     1     1     1     1     1 

T 

r; 

C<)  t^  CO  CN  GO 

CO  d  CO  1-H  d  o 
c:  ^  -^  -^  oo 

CO 

01  01  -O  CO  Ci  1-1 
,-1  1-1  (M 

1  +  1  i  k  + 

05 
1 

a 

r- 

-*  t^  lO  p  p  p 

d  d  o^i  CO  lO  c-i 

C~w  lO  TjH  -*  00  i-i 

c 

T-H  ^  CO  t-;  p  p 

■  rH  d  CO  I*  C4 

1— 1  r-l           i-< 

+  +  +  +  +  + 

+ 

H 

s 

CN  CO  p  -^  0O_  p 

d  d  d  CO  T-i  d 

O  vC  ^  -^  OO  0^ 

CO 

CI  p  t-;  p  -*  p 

'  t-^  1-H  CO  d  lo 

<M  CO  CO  •* 
++++++ 

CO 
+ 

^5 

o 

s 

W 

•J 

>5 

1 

o  o  o  -^  o  o 

H  flH  plH  -<  PH  P4 

0) 

to 

R,^   rt   o   "^   «* 
q    ^^    !h  ^    !h    ^ 

o  o  o  -^  o  o 
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0) 

> 

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•pailildv!  S31UH  001  Jad 

•paSpnC  aDuujsiQ  }o  aojag  JO 
•juda  aoj 

•SNij  ^o  aaa 

I\[Q 

sj  JO  sxnaiMoanf 

OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


37 


lO  t~-M 

C-.  CO 

5^ 

c;  t-  t^ 

OJ  CO 

l^ 

C:  O  o 

-#  o 

CO 

00  c:  1^  O  O 
c;  r~  t^  «  ■* 

.-H    OO    (Tl    Tt<    T-H 

(M 

aa  t~  iS 

C  t—  t:- 

:s  CO 

o 

C  t~-  O  -M  O 


oc  -M  t-  rr  X  i^] 


CC  I-;  CC  C;  CO 

X  c;  x'  r:  "' 
Ci  X  t—  w  o 


c:  r^  cr  r-;  o 
t-^  :f  ri  x'  ^ 


X  ri  r-;  O  cc 
t-^  i~^  t-^  '>!  c; 

C;  t:~  l^  O  iM 


^  -M  ut;  X  O 


O  p  cc  «  O 
CO  ^  O  -1"  00 
C.  Xj  t^  '^  ^r 


+  +  +     I     + 


CO  lO  O  CI  o 

*  r-5  CN  1-i  i-H 

+  +  +  I  + 


C   .-H    -M  r-H  CO 

1     1  ++   1 

1 

■*  30  lyj  t-; 

+++ 1 

+ 

.-H  :c  --r>  o  o 

■   r-H  >-;  to 

1  1  +  1  1 

7 

cc  -M  C^  IM  O 

1  + 1  +T 

o 

CO 

1 

.-;  c^  Ofl  p  CO 

T-H        '   cc   I-<   t-^ 

1  ++++ 

+ 

^.^  o  cc  c:  c<i  lO 

2         '       '  CO  r-<  O 

2,+  +  +  +  + 

+ 

i-  t--  •>q  cc  cc 

++++  1 

+ 

O  -*  Ct   p  ^ 
r-;  1-!        '  1-^    X 

1  1  +  1  T 

1 

p  r-;  lO  lO  ■^_ 

1     1   +   1   + 

1 

CO  CO  !M_  X 
O  t~^  ■*  O"  '^' 


+ 


■^  P  '-^  ■"! 

O  t-^  -^  O  (N 

o  ci  ci  c:  n 


o      o 

O  O  C:  O  O 

o  o  c;  o  o 


cc  (N  cc  X 


cc  c  o 
o  X  cc  uc  ^ 


cq    O  '■ 


CO  O  tH  CO 


r~  cc  t—  O 

O  c:  CO  c:  X 


IM  CO  :0  O 


O  i_C  w  ■M 


X  o  i~  -^ 


3b' ^ 
a  u 

o  o 


rt  o  =5  5 
=  ^  o  c 

^  <   t,  &L< 


<D  ?  =  S  =  S 

ii    i)    U    2    tH    ^ 
C   5-1   ^H    =;   ^H   tH 


S    S    <U    fi    fi 

2  S  n  s  t:  n 

r^^  c^  O  c3  rf 
G    tH    u    >i    t<    t-l 

c  o  o  ■^  o  o 

E-i  fs,  flH  ■<  PH  pIH 


'paSpnt  a3a6)si0  jo  jcujj  jo 

•JUSDJ.IJ 


-p3I|ddB    33U11)   00[   J9d 

Xi)3^03  paSpiiC  saoii)  -ojj 


•aoKVxsiQ  JO  sxxajvoajp 


JO  sxNaKoaap 


38 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


o 

OS 

^  a 
CO  ^ 


o 


» 

K 

H 

1^ 

e 

H 

< 

o 

;?; 

r« 

m 

<! 

ttl 

ce 

s 

^ 

H 

-^ 

a 

is! 

« 

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^o 

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2o 

lO 

Ah5 

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"^ 

XH 

•* 

«Ph 

1^, 

D 

H 

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CO 

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o 

« 

mtH 

H 

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S 

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B 

!^ 

00 

c^ 

O 

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H 

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lO 

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J 

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K 

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M 

a 

^ 

OQ 

pj 

H 

g 

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c 

c 

o 
<A 

< 

iffl  l^  CO  C5  QO 


T-H  GC  iM  •*  ' 


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oi  Ot  -^  X  CO 

c;  1—  i^  >o  -^ 


O  ■*  l<!  CO  :0 
05  CO  c:  c:  CO 

C-.  cs  :C  :^  CO 


(M  CO  00  CO  r~ 


O  (M  CO  >C  O 


(M  fN  o  r-  '>q 

CO  <yj  ci  CO  CO 
c;  CO  ■»  O  -* 


lO 

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CO 

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o 

o 

■* 

■^ 

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^H 

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o 

S    S    S    G    S 

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o  o  o  -5  o  o 
H  6m  tM  •<  Pm  Pm 


•pailddB  sauH}  full  J*<I 


+  +  + 


-*<  X  '^^  o 
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+  +  +  + 


"  f^D  o  n 

I  +  I  + 


1 

1 

+ 

cq  1-1 

-11    + 


CO  o 

I       I       I       I       I 


--I  f>J  CO  »  CO 

'  T-I T-;  d 

t  +++T 


0<I  O  O  CO  lO  > 
■  <n"  >-<  r-^  d 

+  I  +  I    ! 


Ol  C".   Xi  X 

^     '  d  c-j 
I   +   I   + 


+  +  +  +  + 


<r,    CS    a    OJ  g    g 

®  oj  n  g  c  S 

O  O  O  t^  o  o 

H  6h  Pm  <!  Pm  PK 


•paSpnC  aauBisiQ  jo  joaag  jo 
■juao  asj 


•aoNvxsid  JO  sxxaKoanp 


OUE,    NOTIONS    OF    NUMBER    AND    SPACE. 


39 


CO  CO  (M_  00 

O  t-^  -^  <»'  -^ 

O  Ci  C  C5  lO 

CO 
CO 

CO  O  CO  CO 

o  CO  lO  o  t-^ 
o  o;  Ci  c;  "* 

CO 

o  p  O  i-< 
O  I-^  -*  LO  rN 

O  Ci'  c;  o  o 

oi 

CO 

§ 

o 
o 

o 

o 

o 
o 
I— 1 

o 
o 

§ 

2i,        Ci  c:  o 

t-H 

ci 

CO  (^J         o 

o  ci  ci  O  t-^ 
O  Ci  Ci  o  >-o 

CO 
CO 
Oi 

CJ  p  CO  CN 

O  oi  r-  oi  -^ 

O  Oi  Ci  Ci  o 

CO 

8 

CO  >-•:>  o  o 

O  CO  >0  CO  ^ 

O  Ci  Ci  O  Li 

Ol 

CO 

r~  iM  p  T^ 

O  CO   ^  t^.-H 
O  C5  O  CO  lO 

CO 

(M  O  CO  t^ 

o  ci  ri  -^  l-^ 

O  CO  CO  t^  o 

o 

CO 

Tongue 
Forehead 
Forearm 
Abdomen 
Forearm  (a) 
Forearm  (b) 

0) 

ci 
u 
IP 

■paijddo  9.1IUII  oot  Jnd 
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ao  sxx3iv:>a.i 

[* 

- 

EXPERIMENT   C. 


WITH    LINEAL    FIGUKES. 


Apparatus.  —  The  lineal  figures  were  made  from 
cardboard  of  medium  thickness,  but  very  liard  and 
strong.  This  material  was  chosen  to  avoid  temperature 
complications.  Great  care  Avas  taken  that  the  lines, 
and  particularly  the  corners  of  the  figures,  should  be 
perfectly  even,  sharp,  and  accurate  throughout.  The 
importance  of  this  cannot  be  fully  appreciated  unless 
one  has  acted  as  subject  for  a  long  period.  It  is  little 
less  than  marvelous  how  slight  a  cue  will  be  noted  by 
which  to  remember  a  particular  piece  of  apparatus  as 
"  tliat  same  old  one,"  and  so  the  judgments  become 
based  upon  a  fund  of  past  experiences  and  imaginations, 
rather  than  upon  a  new  and  present  impression,  as  is 
absolutely  necessary  for  the  work  here  in  hand.  In  our 
work,  if  any  piece  became  thus  "■  individualized,"  it  was 
at  once  discarded.  The  figures  were  made  like  deep 
pasteboard  boxes,  with  one  end  (that  to  be  pressed  on 
the  skin)  left  open,  as  when  the  lid  of  the  box  is 
off.  The  larger  pieces  were  braced,  as  it  were,  with 
false  bottoms,  one  or  more,  as  needed  to  make  them 
firm. 


OUK    NOTIONS    OF    NUjSIBER    AND    SPACE.  41 

The  figures  used  -were  triangles,  squares,  and  circles. 
The  categories  of  "  distance  "  remained  the  same  as  in 
the  previous  experiments.  There  were  no  longer,  of 
course,  any  ''number"  categories  to  be  observed. 

Method.  —  This  was  precisely  the  same  as  in  Experi- 
ment B,  but  a  new  care  was  required  in  applying  the 
apparatus  to  the  subject.  The  pieces  being  made 
hollow,  like  a  drum,  they  would,  upon  the  least  slipping 
of  the  fingers  over  their  surface  while  handling  them, 
give  out  a  sound  Avith  the  spontaneity  of  a  resonance  box 
or  a  tambourine.  This  sound  would  become  individual- 
ized by  the  subject  for  eacli  particular  piece,  the  same 
as  a  bent  corner  or  an  imperfect  line,  and,  in  a  way 
making  the  judgments  worthless  if  such  sounds  were 
permitted.  The  utmost  care,  therefore,  was  used 
throughout,  in  handling  the  pieces,  to  avoid  every 
particle  of  slipping  or  rubbing  of  the  box,  either  upon 
the  subject's  skin,  or  upon  the  fingers  of  the  operator. 

The  tongue  was  no  longer  investigated,  as  the  appa- 
ratus now  was  too  large  to  work  with  comfortably  in 
the  mouth. 

The  Tables  18a  to  22a,  inclusive,  for  Experiment  C, 
and  18b  to  22b  for  Experiment  D,  will  be  understood, 
after  examination  of  the  similar  ones  for  Experiments 
A  and  B,  without  further  explanation.  These  above- 
numbered  tables  of  Experiments  C  and  D  correspond, 
respectively,  to  Tables  2,  3,  4,  5,  and  10  of  Experiment 


42  OUR    NOTIONS    OF    NUMBER   AND    SPACE. 

A,  and  to  Tables  12,  13,  14,  15,  and  17  of  Experi- 
ment B. 

Table  23.  This  table  shows  the  distribution  of  the 
whole  number  of  figure-judgments.  They  are  calcu- 
lated, in  per  cent.,  from  one  hundred  applications 
of  each  piece  of  apparatus  to  each  of  the  several 
regions  of  skin  worked  upon.  For  example  :  the  three 
numbers,  65.0,  21.0,  14.0,  arranged  vertically  in  the 
upper  left-hand  corner  of  the  table,  mean  that  of  the 
total  number  of  times  that  the  "1  centimeter  "  triangle 
was  applied  to  the  various  regions  of  the  body,  in 
65  per  cent,  of  those  times,  this  triangle  was  judged 
correctly  to  be  a  triangle  ;  in  21  per  cent,  it  was 
misjudged  to  be  a  square ;  and  in  14  per  cent,  a 
circle. 

The  purpose  of  this  table  (to  be  discussed  in  our 
general  study)  is  to  aid  in  comprehending  the  errors 
made  in  judging  the  figures.  , 

The  heary  figures  in  this  table  show  the  correct 
judgments  ;  the  other  figures  show  the  false  judgments. 


EXPERIMENT  D. 


WITH    SOLID    FIGURES. 


Apparatus.  —  Like  that  for  Experiment  A,  except 
that  the  pieces  were  made  of  cork,  and  solid 
throughout. 

Method.  —  Precisely  that  of  Experiment  C. 

Tables. — See  ''The  Tables'^  (page  41)  under  Experi- 
ment C. 


44  OUR   NOTIONS    OF    NTJMBER   AND    SPACE. 


Table  18a.     Hxperiment  C.  —  With  lineal  figures. 


FOREHEAD. 


Person 

Distance 

(Centi- 
meters) 

Averages  of  N.  and  P. 

Figure 

Triangles 

Squares 

Circles 

Averages 

of 

T.,S.andC. 

O 

< 
H 

m 

No.  times  judged  cor- 
rectly per  100  times 
applied. 

1 

1.5 

2 

2.5 

3 

3.5 

86 
52 
56 
50 
42 
48 

(226) 

32 
50 
32 
30 
44 
88 

72 
54 
64 
38 
48 
62 

63.3 
52.0 
50.7 
39.3 
44.7 
66.0 

En 
O 

H 

Averages 

55.7 

46.0 

56.3 

52.7 

Per  cent. 

of  Error  of  Distance 

judged. 

1 

1.5 

2 

2.5 

3 

3.5 

+  7.0 
-4.0 
+    .5 
-  1.6 
-1.0 
-8.2 

(229) 

+  42.0 
+  20.0 
+  28.5 
+  17.6 
+  15.3 
—    1.7 

+  16.0 
+  12.6 
+     5.0 

+    4.8 

-  .6 

-  6.7 

+  21.7 
+    9.5 
+  ,11.3 
+    6.9 
+    4.6 
-    5.2 

Averages 

-  1.3 

+  20.3 

+    5.2 

+    8.1 

Cm 
O 

Q 
•-5 

No.  of   correct  judg- 
ments per  100  times 
applied. 

1 
1.5 

2 

2.5 

3 

3.5 

74 

70 
68 
60 
72 
54 

(232) 

32 
46 
64 

72 
78 
82 

54 

38 
54 

86 
80 
84 

Averages 

66.3 

62.3 

66.0 

64.87 

OUR    NOTIONS    OF   NUMBER    AND    SPACE. 


45 


Table  18b.     Experiment  D.  —  "With  solid  figures. 


FOREHEAD. 


Person 

Distance 

(Centi- 
meters) 

Averages  of  N.  axd  P. 

FiGCRE 

Triangles 

Squares 

Circles 

Averages 

of 

T.,S.andC. 

o 

H 

No.  times  judged  cor- 
rectly per  100  times 
applied. 

1 

1.5 

2 

2.5 

3 

3.5 

52 
52 
22 
22 
34 
84 

(235) 

34 
30 
26 

28 
28 
80 

56 
44 

40 
14 
42 

78 

47.3 
42.0 
29.3 
21.3 
34.7 
80.7 

O 

H 

o 

a 

Averages 

44.3 

37.7 

45.7 

42.6 

Per  cent. 

of  Error  of  Distance 

judged. 

1 

1.5 

2 

2.5 

3 

3.5 

+  32.0 
+   2.0 
+  22.0 
+  11.6 
+   6.3 
-  2.8 

(238) 

+  53.0 
+38.5 
+  40.0 
+  21.2 
+   8.3 
—  3.4 

+  29.0 
+  25.3 
-17.0 
+  12.8 
+   3.0 
-  3.1 

+  38.0 
+  21.9 
+  26.3 
+  15.2 
+   5.9 
-  3.1 

Averages 

+  11.8 

+  26.3 

+  14.0 

+  17.4 

>£4 

o 

No.  of  correct   judg- 
ments per  100  times 
applied. 

1 

1.5 

2 

2.5 

3 

3.5 

74 

68 
62 
76 

82 
88 

(241) 

40 
58 
68 
82 
92 
94 

54 

6(5 
84 
82 
86 
94 

Averages 

75.0 

72.3 

77.7 

75.0 

46 


OUE    NOTIONS    OF   NUMBER    AND    SPACE. 


Table  19a.     Experiment  C.  —  With  lineal  figures. 


FOREARM. 


Person 

Distance 

(Centi- 
meters) 

AVERAGE.S  OF  N.   AND  P. 

FiGUKE 

Triangles 

Squares 

Circles 

Averages 

of 

T.,S.  andC. 

H 

No.  times  judged  cor- 
rectly per  100  times 
applied. 

1 

1.5 
2 

2.5 

3 

3.5 

44 

36 
40 
38 
20 
12 

(244) 

32 

28 
38 
26 
38 
76 

30 
30 
42 
32 

48 
28 

35.3 
31.3 
40.0 
32.0 
35.3 
38.7 

\n 

Averages 

31.6 

39.6 

35.0 

35.4 

H 

a 

Per  cent. 

of  Error  of  Distance 

judged. 

1 

1.5 

2 

2.5 

3 

3.5 

+  40.0 
+  24.6 
+  11.0 

-  1.6 
-18.7 

-  23.4 

(247) 

+  54.0 
+  34.0 

+  22.5 
+    9.2 
+    3.7 
-    4.0 

+  71.0 
+  12.7 
+  10.0 

-  5.2 

-  7.3 

-  18.0 

+  55.0 
+  23.8 
+»14.5 
+    1.5 

-  7.4 

-  15.4 

Averages 

+    5.3 

+  10.7     1   +  10.9 

+  12.0 

o 

No.  of  correct  judg- 
ments per  100  times 
applied. 

1 

1.5 

2 

2.5 

3 

3.5 

62 
58 
68 
44 
38 
26 

(250) 

56 
48 
48 
54 
56 
76 

40 
44 

58 
48 
60 
82 

Averages 

49.3 

56.3 

55.3 

53.6 

OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


47 


Table  19b.     Experiment  D.  —  With  solid  figures. 


FOREARM. 


Person 

Distances 

(Centi- 
meters) 

Averages  of  N.  and  P. 

Figure 

Triangles 

Squares 

Circles 

Averages 

of 

T.,S.andC. 

w 
< 

No.  times  judged  cor- 
rectly per  100  times 
applied. 

1 

1.5 

2 

2.5 

3 

3.5 

38 
32 
38 
14 
58 
38 

(253) 

30 
30 
32 
24 
50 
80 

40 
26 
30 
32 
54 
80 

36.0 
31.3 
33.3 
23.3 
54.0 
66.0 

U* 

Averages 

36.3 

42.0 

43.7 

40.7 

H 

d 

Q 

Per  cent. 

of  Krror  of  Distance 

judged. 

1 

1.5 
2 
2.5 

3.5 

+  53.0 
+    9.3 
+  14.0 

-  9.2 

-  6.0 

-  13.4 

(256) 

+  65.0 
+  36.7 
+  22.5 
+  26.8 
+    6.3 
-    4.0 

+  52.0 
+  26.6 
+    4.0 
4-    8.8 

-  2.0 

-  3.4 

+  56.7 
+  24.2 
+  13.5 
+    8.8 

—  .6 

—  6.9 

Averages 

+   7.9 

-f-25.5 

+  14.3 

+  15.9 

o 
a  2 

O  Cm 

No.  of  correct  judg- 
ments per  100  times 
applied. 

1 
1.5 

2 
2.5 

3.5 

52 
56 
44 
36 
(i4 
54 

(259) 

36 
44 

38 
38 
68 
68 

42 

56 
(i6 
62 
80 
88 

Averages 

51.0 

48.7 

65.7 

55.1 

48 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


Table  20a.     Experiment  C. — "With  lineal  figures. 


ABDOMEN. 


Person 

Distances 

(Centi- 
meters) 

Averages  of  N.  and  P. 

Figure 

Triangles 

Squares 

Circles 

Averages 

of 

T.,S.andC. 

< 

H 

0 

No.  times  judged  cor- 
rectly per  100  times 
applied. 

1 

1.5 

2 

2.5 

3 

3.5 

50 
36 
36 
30 
30 
36 

(262) 

48 
32 
26 
40 
40 
76 

42 
46 
32 
42 
40 
34 

46.7 
38.0 
31.3 
37.3 

36.7 

48.7 

.    h 

Averages 

36.3 

43.7 

39.3 

39.8 

a 
o 

0 

Per  cent. 

of  Error  of  Distance 

judged. 

1 

1.5 

2 

2.5 

3 

3.5 

+  43.0 
+  19.3 

-  1.5 

-  3.6 

-  10.0 

-  17.7 

(265) 

+  44.0 
+  17.3 
+  28.5 
+  13.2 
+    6.6 
-    5.4 

+  52.0 
+  18.6 
+    1.5 
+      .8 

-  3.0 

-  13.1 

+  46.3 
+  18.4 
+  '  9.5 
+    3.5 

-  2.1 

-  12.1 

Averages 

+    4.9 

+  17.4 

+    9.4 

+  10.6 

O 
1-5 

No.  times  judged  cor- 
rectly per  100  times 
applied. 

1 

1.5 

2 

2.5 

3 

3.5 

52 
72 
48 
74 
56 
46 

(268) 

38 
52 
46 
60 

72 
78 

36 
64 

86 
82 
92 
80 

Averages 

58.0 

57.7 

73.3 

63.0 

OIJR    NOTIONS    OF    NUMBER    AND    SPACE. 


49 


Table  20b.     Experiment  D.  —  With  solid  figures. 


ABDOMEN. 


Peksox 

Distance 

(Centi- 
meters) 

Averages  of  N.  a:sd  P. 

Figure 

Triangles 

Squares 

Circles 

Averages 

of 

T.,S.andC. 

o 

15 

■< 
H 
to 

No.  times  judged  cor- 
rectly per  100  times 
applied. 

1 

1.5 

'> 

2.5 

3 

3.5 

34 
24 

54 
26 
32 
48 

(271) 

44 

28 
30 
32 
38 
74 

46 
32 
52 
34 

48 
68 

41.3 
28.0 
45.3 
30.7 
39.3 
63.3 

O 

o 
s 

Averages 

36.3 

41.0 

46.7 

41.3 

Per  cent. 

of  Error  of  Distance 

judged. 

1 

1.5 

2 

2.5 

3 

3.5 

+  55.0 
+     .6 
+  2.5 
-12.0 
-  5.3 
-15.7 

(274) 

+  48.0 
+  26.0 
+  27.0 
+  11.2 
+  6.3 
-  4.9 

+  36.0 
+  26.6 

-  1.5 

-  2.0 
-10.0 

-  6.2 

+46.3 
+  17.7 
+  9.3 

-  .9 

-  3.0 

-  8.9 

Averages 

■+  4.2 

+  19.0 

+   7.1 

+  10.1 

64 

o 

s  2 

3 
1-5 

No.  tinies  judged  cor- 
rectly per  100  times 
applied. 

1 

1.5 

2 

2.5 

3 

3.5 

58 
62 
50 
68 
66 
52 

(277) 

34 
42 
46 
62 

m 

72 

46 
42 
62 
70 
72 
90 

Averages 

59.3 

53.7 

63.7 

58.9 

50 


OUR    NOTIONS    OF   NUMBER    AND    SPACE. 


Table  21a.     Experiment  C.  —  "With  lineal  figures. 


(a)  FOREARM. 
Pressing  evenly  three  times  only. 


Person 

Distance 

(Centi- 
meters) 

Averages  of  N.  axd  P. 

Figure 

Triangles 

Squares 

Circles 

Averages 

of 

T.,S.andC. 

< 

p 

No.  times  judged  cor- 
rectly per  100  times 
applied. 

1 

1.5 

2 

2.5 

3 

3.5 

54 
30 
48 
24 
24 
22 

(280) 

30 
26 
36 
14 
52 
54 

30 
26 
50 
16 
44 
22 

38. 

27.3 

44.7 

18.0 

40. 

32.7 

p^ 

Averages 

33.7 

35.3 

31.3 

33.4 

03 

H 

o 
p 

1-3 

CS 
jj.    to 

::  O.I' 

o 

1 

1.5 

2 

2.5 

3 

3.5 

+  41.0 
+  12.7 
+    3.5 

-  7.0 

-  13.6 
-21.4 

(283) 

+  62.0 
+  27.3 
+  21.0 
+  10.8 

-  2.0 

-  3.7 

+  68.0 
+  22.7 
+    7.5 
+    4.8 

—  0.6 

-  18.0 

+  57.0 
+  *0.9 
+  10.7 
+    2.7 

-  7.3 

-  14.4 

Averages 

+    2.4 

+  19.2 

+  13.1 

+  11.6 

(4 
O 

T 

•r    a. 

'S  2  -d 
o   g  1. 
c  -S 

.  s 

o   S 
'A 

1 

1.5 

2 
2.5 

3.5 

72 
00 
54 
52 
50 
39 

(286) 

36 

38 
96 
50 
60 
60 

48 
50 
52 
56 
52 
66 

Averages 

53.0 

50.0 

54.0 

52.3 

V  ■     -      -     ' 

OUll    NOTIONS    Or^  NUMBER   Aim.  SPACE.  51 

-■  -if:  ■  >' 


Table  21b.     Experiment  D.  —  With  solid  figures. 


(a)  FOREARM. 
Pressing  evenly  three  times  only. 


Pkrsox 

Averages 

OF  N.  AXD 

P. 

Distances 
(Centi- 

Averages 

Figure 

meters) 

Triangles 

Squares 

Circles 

of 
T.,S.andC. 

H 

(289) 

^5 

1 

54 

44 

22 

40.0 

^S^' 

1.5 

23 

22 

24 

23.0 

H 
O 

2 

30 

18 

32 

26.7 

Z 

<o    ^  ~ 

2.5 

48 

26 

30 

34.7 

E- 

B  >," 

3 

32 

52 

50 

44.7 

P 

o   « 

3.5 

20 

80 

46 

48.7 

b 

Averages 

34.5 

40.3 

34.0 

36.3 

(292) 

1^ 

1 

+  4G.0 

+  49.0 

+  64.0 

+  53.0 

S  ^  -3 

1.5 

+    9.3 

+  40.6 

+  31.3 

+  27.1 

O    O    M 

2 

+    8.0 

+  33.0 

+  21.5 

+  20.8 

>-s 

3 

2.5 

-    1.0 

+  14.4 

+    7.2 

+    6.7 

3 

-11.0 

+    8.3 

-    5.3 

-    2.7 

o 

3.5 

-1G.3 

-    4.0 

-    3.4 

-    7.9 

Averages 

+    5.7 

+  23.5 

+  19.2 

+  16.1 

Eiu 

"2  2 

(295) 

O 

"5 

1 

46 

44 

40 

«  2  -s' 

1.5 

42 

36 

32 

« r; 

t-   u  -; 

2 

40 

28 

44 

s  2 

b^S 

2.5 

40 

30 

52 

®  a 

o 

48 

50 

36 

>-i 

3.5 

24 

GG 

60 

Averages 

41.0 

44.3 

44.0 

43.1 

52 


OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


Table  22a.     Experiments  C  and  D. — Witli  lineal  and  solid 
figures. 

SUMMARY.  —  Averages  brought  forward  from  Tables  18a 
to  21b. 


Person. 

Average  of  N.  and  P. 

Tri- 
angles 

Averages 

Figure. 

Squares 

Circles 

of 

T.  S.  and  C. 

>% 

(298) 

1   -d 

Forehea..       j  ^^f  > 

55.7 
44.3 

46.0 
37.7 

56.3 
45.7 

52.7 
42.6 

S  a 

T^                     I  Lineal 

31.0 

39.6 

35.0 

35.4 

:  ^ 

Forearm         j  g^^.^^ 

36.3 

42.0 

43.7 

40.7 

CD     CC 

T^               ,  s  (  Lineal 

33.7 

35.3 

31.3 

33.4 

3  2 

Forearm  («)  {  g^^j^^ 

34.5 

40.3 

34.0 

36.3 

m 

All                 (  Lineal 
Abdomen       j  g^,j^ 

36.3 

43.7 

39.3 

39.8 

% 

^  2 

36.3 

41.0 

46.7 

41.3 

H 

.                      (  Lineal 
Averages       j  g^jj^ 

39.325 

41.15 

40.47 

40.30 

fi 

o 
"A 

37.850 

40.25 

42.52 

40.26 

U 

o 

Average  of  Lineal  and  Solid 

38.590 

40.70 

41.50 

40.28 

(301) 

Z 

m 

Forehead       i^^f 

-   1.3 

+  20.3 

+    5.2 

+   8.1 

S 

S  2 

+  11.8 

+  26.3 

+  14.0 

+  17.4 

t 

2  s 

,.                     (  Lineal 
iorearm        j  g^jj^ 

+   5.3 

+  19.7 

+  10.9 

+  12.0 

w  !• 

+   7.9 

+  25.5 

+  14.3 

+  15.9 

■s^ 

,,               /  >  I  Lineal 
iorearm(a)Jg_^jj^^ 

+   2.4 

+  19.2 

+  13.1 

+  11.6 

^  « 

+   5.7 

+  23.5 

+  19.2 

+  16.1 

^1 

Abdomen       \^^^-' 

+   4.9 
+   4.2 

+  17.4 
+  19.0 

+   9.4 
+   7.1 

>+10.6 
+  10.1 

Averages       j  ^^T^ 

+  2.8 

+  19.1 

+   9.6 

+  10.5 

o 

+   7.4 

+  23.6 

+  13.6 

+  14.9 

Average  of  Lineal  and  Solid 

+   5.1 

+  21.3 

+  11.6 

+  12.7 

to 

(304) 

a    . 

a  .s 

-r^      ,       1        (  Lineal 
Forehead       -^  „  1  •  1 
(  Solid 

()6.3 
75.0 

62.3 
72.3 

66.0 

77.7 

64.9 
75.0 

rg'     a 

F„,.ea™,        \^^ 

49.3 

56.3 

55.3 

53.6 

)^ 

.ai* 

51.0 

48.7 

65.7 

53.1 

o 

.,      CO 

^               ,  .  (  Lineal 

53.0 

50.0 

54.0 

52.3 

to 

ll 

Forearm  («)  j  g^j.^^ 

41.0 

44.3 

44.0 

43.1 

[2J 

9.  ^ 

.  1  ,                 (  Lineal 

58.0 

57.7 

73.3 

63.0 

1 
o    u 

0    " 

Abdomen       }  g^j.^^ 

59.3 

53.7 

63.7 

58.9 

t-5 

Averages       { ^^^^^ 

56.65 

56.57 

56.57 
54.75 

62.15 

62.77 

58.45 
58.025 

Average  of  Lineal  and  Solid 

56.61 

55.66 

62.46 

58.20 

OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


53 


(C     ^ 


X 

3c 

s 

(>i  cri  o  o  oi  «D  ci  1-; 

X 

ad  t-^  (m'  o  r-i  to  o  c" 

+  + 

ci 

+ 

H 
CM 

2 

ei 
CO 
H 

2 

CO 
IN 

O  O  00  -O  -N  ac  3D  r: 

o  X  rr  o  CO  ^  T}i  o 

to  •* 

bS 

0<)  i-H  -^  p  ■^_  p  r-H  p 
lO  CC  O  O  -^  t-^  "I^  CO 

1  1  T  1  T  1  T  1 

X  t- 

,-;  to 

1         1 

OS 
1 

t-;  t-;  CO  p  O  t-;  t-;  or 

•<1"  -*  ut"  ^  o  '^  l5  c: 
■<*i  cc  OT  u"  -*  •<*  :c  'T 

5:  ^ 

pc:^wcct— r-c; 
■*  o  t~^     '  t-^  '>5  ?ci  CC 

+  +  1    1    1    M    1 

T  1 

p 

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o 

^  CT  ??  O  0?  p  t-;  or  r- 
o  c:  r4  •m'  n  X.  Tf  I—'  o 

^  ec  ?q  r:  ?q  .-(  ct  ct  cc 

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2    O  id  .-<■  CO  CN  O  CO 

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+  + 

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t—  ^T  O  "*  t^  1^  CC  tC 

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O  •M  TP  ct  -*  I-J  cc  ■* 

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CO  CC  L.C  LC  t-;  CO  O  CC 

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C:  --I  CC  •^'  O  t-^  X  i>^ 
c^  CI  CI  c^  cq  .-^  T-i 

++++++++ 

X  ji 

+  + 

+ 

0 
a 

OJ 

to 

(C 

)H 

0) 

> 

< 

cocoorooot^co 
c?  t-^  o  :2  CO  o  o  ^ 

CO  iM 

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t-J  X  o  :c  t-^  CC  o  to' 

CI  CC  LC  LC  uC  lC  -*  Tf 

++++++++ 

p  o 
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+  + 

+ 

"5  —  "?  —  "^^ "?-- 

1-3  X  .^  X  '-2  X  f2  X 

-^      ^      "^     c 
<s      S      S      ^ 
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^       rt       ce       o 
o      o      o      -i; 

>-.            Jh            Sh            - 

c        o        o       -^ 
fc^      £=^      t.      <1 

a 
u 

0) 

> 

< 

hi 
> 

i3x 

2 

0 

-3 
3 

1 

-< 

K 

< 

'i      3      3      1 

1*                *-!                '-I                 3 

d         c3         rt         5 
<U           O)           <U          .-i 

S      o      S      ^ 

ph    pR    p^    <; 

32 
> 

•siuaniSpnp  aotreisia  Jo 

JOJJa   JO    -^UaD   J3d^ 

HJKV 

xbi(j; 

j< 

>    sixaKyuax' 

54 


OUR    NOTIONS    OF    NUMBER   AND    SPACE. 


Table  23.     Experiments  C  and  D  — 


(For  explanation  of  this 


DIST.US'CES. 

1 

1.5 

■2 

Figures. 

T. 

S. 

c. 

T. 

S. 

C. 

T. 

s. 

c. 

(307) 

(308) 

(309) 

Exp.  C. 

Triangle 

65.0 

41.0 

24.5 

65.0 

36.0 

2G.0 

59.5 

.32.0 

13.5 

Square 

21.0 

40.5 

31.0 

28.0 

46.0 

25.0 

32.5 

53.5 

24.0 

Circle 

14.0 

18.5 

44.5 

7.0 

18.0 

49.0 

8.0 

14.5 

62.5 

(014) 

(315) 

(316) 

Exp.  D. 

Triangle 

57.5 

45.5 

24.5 

57.0 

35.0 

25.5 

50.5 

,33.5  20.0 

Square 

27.0 

38.5 

30.0 

27.5 

45.0 

25.5 

32.5 

45.0  16.0 

Circle 

15.5 

10.0 

45.5 

15.5 

20.0 

49.0 

17.0 

21.5'' 64.0 

\ 


OUR    NOTIONS    OF    NUJklBEE    AND    SPACE. 


55 


With   lineal   and   solid   figures. 


table,  see  page  42.) 


2.5 

3 

3.5 

Averages 
of  all  Distances. 

Total 

Aver- 

T. 

s. 

C. 

T. 

S. 

c. 

T. 

S. 

C. 

T. 

S. 

c. 

ages. 

(310) 

(311) 

(312) 

(313) 

57.5 
32.5 
10.0 

21.5 
59.0 
19.5 

4.0 
28.0 
68.0 

54.o'  16.0 

38.0  66.5 

8.0  17.5 

11.5 
17.5 
71.0 

39.012.5'    5.0 
44.0  74.0  17.0 
17.0  13.5  68.0 

56.6 

32.6 
10.6 

26.5 
56.6 
16.9 

14.1 
23.7 
62.1 

L58.5 

(317) 

(318) 

(319) 

(.320) 

55.0 

28.5 
16.5 

30.5 
45.0 
24.5 

14.5 
19.0 
66.5 

65.0 

31.0 
4.0 

13.0 
70.5 
16.5 

10.5 
21.0 
68.5 

54.5 

38.0 
7.5 

19.0 

75.0 

6.0 

5.0 
12.0 
83.0 

56.6 

.30.7 
12.6 

29.4 
53.1 
17.4 

16.6 
20.6 
62.7 

1 


V    ^ 


EXPERIMENT   E. 


WITH    A    MOVIXG    PENCIL. 


At  a  certain  stage  in  our  studies,  we  shall  have  to 
inquire  wliat  part  each  of  several  elements,  which  we 
know  enter  into  the  formation  of  every  judgment, 
individually  plays.  Important  among  these  are  "mass,"' 
both  of  stimulations  and  of  feelings  ;  ''intensity,"  both 
peripheral  and  central ;  the  ''  time  rate  "  of  stimulation, 
and  of  mental  response.  Particularly  we  shall  wish  to 
know  the  separate  influence  of  each  of  these  factors,  in 
order  to  comprehend  their  united  action  in  producing  a 
judgment  as  a  whole.  Experiment  E,  still  pursuing 
the  comparative  method  of  investigation,  has  this 
exigency  in  view.  It  differs  from  all  the  foregoing 
experiments  by  introducing  motion  over  the  skin. 

The  apparatus  and  the  method  were  of  the  simplest 
kinds.  The  pencil  was  of  ivory,  5  millimeters  in  diam- 
eter, and  rounded  hemispherically  at  the  end.  It  was 
always  kept  at  the  skin  temperature,  was  held  vertical, 
and  applied  by  an  assistant.  The  region  to  be  worked 
upon  was  laid  out  in  squares  by  dots  of  ink  one  centi- 
meter apart.  The  pencil  was  always  drawn  in  the  same 
direction,  i.e.,  horizontally  on  the  forehead,    and  down 


58  Oim    NOTIONS    OF    NUMBER    AND    SPACE. 

on  the  forearm  and  the  abdomen.  Four  categories  of 
motion  were  investigated :  Quick  and  Heavy ;  Quick  and 
Light ;  Slow  and  Heavy  ;  Slow  and  Light.  The  "quick" 
and  "  slow  "  movements  were  timed  by  a  metronome, 
until  we  had,  by  continued  habit,  well  acquired  the  beat. 
''Quick"  was  at  the  rate  of  about  20  centimeters  per 
second  ;  and  slow  about  2  centimeters  per  second.  No 
attempt  was  made  to  gauge  the  pressure  exactly. 
"  Heavy  "  was  as  liard  as  could  be  borne  for  a  length  of 
time  without  pain.  "Light"  was  as  light  as  could  be 
distinctly  and  evenly  felt. 

The  tables  are  so  like  the  other  tables  that  little 
further  explanation  will  be  needed.  As  only  two  sub- 
jects Avere  available  at  the  time  the  experiment  was 
performed,  a  double  number  of  applications  was  made, 
and  each  person  and  their  results  kept  separate,  as  is 
shown  in  the  tables. 


OUR    NOTIONS    OF    Nr:MBER    AND    SPACE. 


50 


Table  24.     Experiment  E.  —  "With  a  moving  pencil. 


FOREHEAD. 


Peksox. 

Distance 

(Centi- 
meters) 

AVEKAGE   OF  N. 

AJJD  P. 

Mode  of 
Motion. 

Quick 
and 
Light 

Quick 

and 
Heavy 

Slow 
and 
Light 

Slow 

and 

Heavj' 

Averages 

O    g 

(323) 

•d  ~ 

1 

96 

96 

94 

92 

94.5 

udge 

loot 

lied. 

2 

82 

86 

80 

80 

82.0 

H 

3 

74 

82 

80 

76 

78.0 

is 

s  s.| 

4 

56 

64 

72 

62 

63.5 

l>: 

5 

68 

78 

76 

56 

69.5 

1^ 

6 

90 

94 

90, 

94 

92.5 

Averages 

77.67 

83.33 

82.0 

76.67 

80.0 

rr. 

(326) 

'A 

3! 

1 

-1-  4.0 

+  4.0 

+  6.0 

+  8.0 

+  5.5 

■e  'S  -• 

2 

-1-  7.0 

+  1.0 

+  4.0 

+  8.0 

+  1.5 

0 

5  <"    iC 

3 

-4.6 

0 

+  1.3 

+  7.3 

+  1.0 

•-5 

4 

-  6.5 

+  6.0 

-1.0 

+  6.0 

+  1.1 

t-^    C 

5 

-  1.6 

+  1.2 

+  2.4 

+  0.4 

+  2.1 

c 

6 

-2.0 

-1.0 

-1.6 

-1.0 

-  1.4 

Averages 

-2.9 

+  1.9 

+  1.8 

+  5.8 

+  1.6 

60 


OUE    NOTIONS    OF    NUMBEK,   AND    SPACE. 


Table  25.     Experiment  E.  —  "With  a  moving  pencil. 


FOREARM. 


Pebson. 

Distance 

(Centi- 
meters) 

Average  of  N. 

AND   P. 

MODK  OF 

Motion. 

Quick 
and 
Light 

Quick 

and 
Heavy 

Slow 
and 
Light 

Slow 

and 

Heavy 

Averages 

8  I 

(331) 

•^■9 

1 

91 

88 

74 

61 

78.5 

■^s-g 

2 

55 

77 

53 

52 

59.25 

H 

3 

60 

56 

48 

47 

52.75 

S5 

gl| 

4 

38 

46 

39 

35 

39.5 

H 

g.^ 

5 

50 

46 

42 

43 

45.25 

P 
O 
H 

1^ 

6      . 

42 

80 

29 

63 

55.0 

Averages 

56.0 

66.5         47.5 

50.2 

55.0 

(336) 

1 

+   9.0 

+  13.0 

+  27.0 

+  47.0 

+  24.0 

a  5  ■«■ 

2 

-  8.5 

+   7.5 

+   7.0 

+  21.5 

+  6.9 

K> 

"   o  .5? 

3 

-10.0 

+  11.0 

+  2.6 

+  15.5 

'+  4.8 

Hs 

<U    u    3 

4 

-  8.2 

+  5.5 

-  3.5 

+  8.2 

+     .5 

Ch   o  -^ 

5 

-  8.8 

+  2.2 

-  9.2 

+   1.0 

-  3.9 

o 

6 

-13.8 

-  3.0 

-16.2 

-  9.1 

-10.3 

Averages 

—  6.7 

+  6.0 

+   1.3 

+  14.0 

+  3.7 

OUR    NOTIONS    OF    NU:MBER    AND    SPACE. 


61 


Table  26.     Experiment  E.  —  With  a  moving  pencil. 


ABDOMEN. 


Person. 

Distance 

(Centi- 
meters) 

Average  of  N. 

AND  P. 

Mode  of 

MOTIOX. 

Quick 
and 
Light 

Quick 

and 
Heavy 

Slow 
and 
Light 

Slow 

and 

Heavy 

Averages 

8  s 

(341) 

r3  ~ 

1 

69 

63 

47 

49 

57.0 

•Is-g 

2 

66 

72 

49 

51 

59.5 

u 

d  °  .s 

3 

53 

52 

45 

48 

49.5 

S5 

sl| 

4 

32 

43 

39 

39 

38.25 

H 

s  ^  * 

5 

43 

39 

39 

44 

41.25 

0 

05 
H 

No.  t 
rect 

6 

44 

62 

39 

61 

51.5 

Averages 

51.17 

55.17 

42.98 

48.67 

49.5 

0 

(346) 

c 

03 

1 

+  31.0 

+  30.0 

+  59.0 

+  58.0 

+46.7 

■s  2  -5 

2 

+   9.0 

+  10.5 

+24.5 

+  26.0 

+  17.5 

"  0  ^ 

3 

+  2.0 

+   7.3 

+  8.3 

+  15.0 

+  8.1 

•-» 

,?  0  -^ 

4 

-  8.2 

+   3.5 

0 

+  5.5 

+     .2 

5 

-11.0 

-  3.4 

-  9.0 

+  2.2 

-  5.3 

w 
"S 

6 

-15.8 

-  8.3 

-15.0 

-  9.3 

-12.1 

Averages 

+   1.2 

+  8.1 

+  11.3 

+16.2 

+  9.2 

62 


OUR    NOTIONS    OF    NUMBEr.    AND    SPACE. 


Table  27.     Experiment  E.  —  With  a  moving  pencil. 


SUMMARY. 
Averages  brought  forw^ard  from  Tables  24,  25  and  26. 


Person. 

Averages  of  N.  ajjd  P. 

JNIODE    OF    IMOTIOX. 

Quick 
and 
Light 

Quick 

and 
Heavy 

Slow 
and 
Light 

Slow 

and 

Heavy 

Averages 

o 

No.  times  judged  cor- 
rectly per  100  times 
applied. 

Forehead 
Forearm 
Abdomen 

77.67 

56.0 

51.17 

83.33 

66.5 

55.17 

(351) 

82.0 
47.5 
42.98 

76.67 

50.2 

48.67 

80.0 
55.0 
49.5 

Averages 

61.6 

68.3 

57.5 

58.5 

61.5 

H 

s 

o 

o 

Per  cent. 

of  Error  of  Distance 

judged. 

Forehead 
Forearm 
Abdomen 

—2.0 
—6.7 
+  1.2 

+  1.9 

+  ().0 
+  8.1 

(356) 

+   1.8 
+   1.3 
+  11.3 

+  5.8 
+  14.0 
+  16.2 

+  1.6 
+3.7 
'+9.2 

Averages 

-2.8 

+  5.3 

+  4.8 

+  12.0 

+4.8 

EXPERIMENT  F. 

COMPARING    HORIZONTAL    AND    VERTICAL    DISTANCE. 

Ajjjjumtus.  —  Fifteen  pieces  of  hard,  thin  cardboard, 
cut  accurately  into  strips  as  follows  :  five  exactly  1 
centimeter  wide,  five  2,  and  five  3  centimeters  wide. 
Then  each  piece  was  again  cut  so  that  one  end  of  each 
of  the  five  cards  of  the  three  distance-categories,  should 
be  respectively,  1,  2,  3,  4  and  5  millimeters  shorter  than 
the  standard  distance,  to  which  the  other  end  of  the  card 
was  originally,  and  still  remains  cut.  It  will  be  seen 
that,  by  applying  first  one  end  and  then  the  other  of 
each  of  these  sets  of  five  cards  in  regular  order,  increas- 
ing ratios  of  difference  between  the  two  distances,  thus 
successively  presented  for  judgment,  Avould  obtain 
throughout  each  set  ;  that  is  for  each  standard  distance. 

Method.  —  The  standard  and  longer  end  of  each  card 
was  always  pressed  on  the  skin  vertically,  that  is  "  up 
and  down  "  of  the  body  or  limb,  wherever  applied,  and 
in  whatever  order  the  two  ends  of  the  card  might  be 
applied.  The  card  was  applied  by  the  assistant  in  a 
manner  like  that  described  for  other  apparatus  in  the 
foregoing  experiments.  The  subject  always  compared 
the  second  application  with  the  Jirsf ;  announcing  that 


64         OUR    NOTIONS    OF    NUIVIBER    AND    SPACE. 

the  second  was  ''  more  "  or  <'  less  "  than  the  first,  or  that 
"no  difference"  could  be  felt. 

As  only  relative  results  were  sought  for,  and  no 
absolute  threshold  distance  was  concerned,  we  neglected 
all  the  "  no  difference  "  judgments.  Only  the  answers 
"  more  "  and  "  less  "  were  recorded  or  coxinted  in  our 
series  of  100  applications. 

Two  categories  of  application  were  used.  In  the  first 
of  these,  marked  always  -^  in  the  left-hand  column 
of  Table  28,  the  order  of  applying  each  card  was,  first 
the  long  end  vertically,  and  then  the  short  end  hori- 
zontally. In  the  second  of  these  direction-categories, 
marked  always  p  in  the  said  column,  the  order  was 
first  the  short  end  of  the  card  horizontally,  then  the  lo7ig 
end  vertically.  As,  therefore,  in  the  first  category  all 
the  answers  ought  to  be  "less,"  and  in  the  second 
category  all  ought  to  be  "  more,"  the  results  of  the  two 
series  ought  to  balance  each  other. 

Table  38  shows  the  results  of  this  Experiment  F. 
The  numbers  expressed  as  fractions  in  the  body  of  the 
table  mean  as  follows  :  the  numerator  gives  the  number 
of  times,  out  of  the  100  applications  of  each  card,  that 
the  second  application  was  judged  to  be  "  more "  than 
the  first.  The  denominator  gives  the  number  of 
times  the  second  application  was  judged  to  be  "  less  " 
than  the  first.  The  numerator  plus  the  denominator 
always  eqvials  100. 


OUR   NOTIONS    OF    NUMBER    AND    SPACE.  05 

Since  in  the  first  line  of  fractions  (see  ^,  Table  28) 
the  longer  end  of  the  card  was  applied  first,  and  then 
the  shorter  end  compared  with  it,  and  in  the  second  line 
(p  in  tables)  the  order  of  applying  the  different  ends  of 
the  cards  was  reversed,  the  shorter  end  being  now  com- 
pared with  the  longer,  plainly  the  sum  of  each  vertical 
pair  of  these  two  horizontal  lines  of  fractions  ought  to 
reduce  to  unity,  and  the  sum  of  the  whole  two  lines  also 
reduce  to  unity,  —  that  is,  if  there  were  no  errors  of 
judgment.  If  every  judgment  was  correct,  all  the  frac- 
tions in  the  first  line  should  read  ^^-q,  thus  recording 
100  "lesses''  and  no  "  ?«o/-es " ;  and  the  second  line 
should  read  all  J-g^.^  or  100  "mores"  and  no  "lesses"  ; 
and  the  sum  of  the  two  lines  should  average  (in  the 
right-hand  or  average  columns)  igg.  Just  in  so  far 
as  the  fractions  depart  from  the  above  formula,  they 
indicate  errors  of  judgment,  and  the  direction  in  which 
they  depart  indicates  the  peculiarity  of  these  errors  ; 
this  is  the  peculiarity  which  we  are  in  search  of.  For 
example,  the  fraction  in  the  upper  left-hand  corner 
shows,  that  of  100  comparisons  of  a  distance  1  centime- 
ter, pressed  on  the  forehead  vertically,  with  a  distance  .9 
centimeters,  pressed  horizontally  immediately  after,  the 
latter  was  judged  correctly  to  be  shorter  than  the  former 
80.75  times,  and  incorrectly  to  be  longer  19.25  times. 
The  parallel  fraction,  next  below,  shows  that  of  100 
applications    of   .9    centimeters    horizontally,    followed 


66  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

by  1  centimeter,  vertically,  the  latter  was  judged  cor- 
rectly to  be  longer  than  the  former  32.5  times,  and 
wrongly  judged  to  be  shorter  67.5  times.  If  we 
now  add  these  two  fractions  we  have  ,VjV>.  ?  and 
tliis  indicates,  in  so  far  as  it  goes,  a  tendency  on  the 
forehead  for  distances  to  seem  longer  when  pressed 
vertically  than  when  pressed  horizontally.  Of  course 
the  sum  totals,  which  foot  up  the  average  columns, 
express  this  tendency  more  generally ;  for  instance, 
the  final  fraction,  for  the  forehead,  shows  for  all  the 
categories  and  degrees  of  difference  worked  with,  that, 
on  the  average,  the  vertical  distances  seem  longer  than 
the  horizontal  ones  by  a  ratio  of  |-§f  |,  or  about  twice 
out  of  three  times. 

Of  course  the  above  ratios  do  not  express  the  amount 
of  error  by  which  the  distance  is  over  judged,  or  fore- 
shortened. Some  idea  of  this  amount  may,  however,  be 
gained  by  casting  the  eye  along  the  line  of  fractions 
from  left  to  right,  while  observing  the  amount  of  differ- 
ence between  the  compared  distances  as  indicated  in  the 
headings  of  the  several  columns.  Thus  it  is  easily  seen 
that  this  difference  for  the  first  column  is  1  millimeter, 
for  the  second  column  2  millimeters,  the  next  3,  the 
next  4,  and  the  next  5,  thus  increasing  from  left  to  right 
through  those  columns  wherein  the  standard  distance 
is  always  1  centimeter.  The  ratios  of  the  first  line  of 
fractions  should  increase,  and  those  of  the  second  line 


OUE    NOTIONS    OF    NUMBER    AND    SPACE.  67 

decrease  from  left  to  right  .through  each  set  of  5 
columns.  What  this  expresses,  in  a  general  way,  under 
the  Psychophysic  Law,  as  to  the  amount  of  error  by 
which  vertical  distances  seem  longer  than  horizontal 
ones,  is  obvious,  though  in  itself  the  amount  cannot, 
from  these  figures,  be  exactly  determined. 


i. 


68 


OUli    NOTIONS    OF    NUMBEH    AND    SPACE. 


Table  28.     Experiment  F.  —  Comparing 


Vertical 
Distances. 

^^  S 

•s  ft 

o 

1            1 

1 

1 

1 

2 

2 

Horizontal 
Distances. 

.9            .8 

.7 

.6 

.5 

1.9 

1.8 

0 

T^ 

19.25 

15.75 

13.25 

14.0 

12.75 

22.75 

26.25 

a 

H 

80.75 

84.25 

86.75 

86.0 

87.25 

87.25 

73.75 

o 

Oi 
> 

H 

32.5 

31.75 

30.5 

39.5 

46.75 

35.25 

44.5 

<1 

V 

67.5 

68.25 

69.5 

61.5 

53.25 

64.75 

55.5 

g 

03 

V 

46.5 

41.5 

30.25 

25.75 

20.75 

66.75 

64.50 

<B 

H 

43.5 

58.5 

69.75 

74.25 

79.25 

33.25 

35.5 

K 

II 

49.25 

47.0 

56.0 

53.5 

55.25 

62.0 

59.5 

^ 

<=5 

r 

50.75 

53.0 

44.0 

46.5 

44.75 

38.0 

40.5 

'i 

CO 

V 

33.25 

33.75 

31.5 

38.0 

31.5 

45.75 

45.5 

'A 
S 

H 

66.75 

66.25 

68.5 

72.0 

68.5 

54.25 

54.5 

H 

> 

H 

27.75 

37.75 

45.75 

33.0 

37.0 

48.25 

43.75 

< 

< 

V 

72.25 

62.25 

54.25 

67.0 

63.0 

51.75 

56.25 

OUR   NOTIONS    OF   NUMBER    AND   SPACE. 


69 


horizontal   and  vertical  distances. 


2 

2 

2 

3 

3 

3 

3 

3 

Totals. 

Sum 
To- 

1.7 

1.6     1.5 

2.9 

2.8 

2.7 

2.G 

2.5 

tals. 

20.75 
79.25 

14.25 
85.75 

12.25 

87.75 

32.0 
68.0 

38.5 
61.5 

32.75 
67.25 

27.5 
72.5 

21.0 
79.0  i 

323 
1177 

1022 

45.25 
54.75 

64.0 
36.0 

61.75 
38.25 

49.75 
50.25 

43.0 

57.0 

56.5 
43.5 

60.0 
40.0 

58.25 
41.75 

699 
801 

1978 

62.5 
37.5 

60.5 

39.5 

48.5 
51.5 

75.25 
24.75 

72.25 

27.75 

82.0 
18.0 

71.0 
29.0 

68.0 
32.0 

836 
664 

1812 

66.5 
33.5 

65.0 
35.0 

68.0 
32.0 

77.0 
23.0 

79.25 
20.75 

77.25 
22.75 

81.25 
18.75 

79.25 
20.75 

976 
524 

1188 

37.5 
62.5 

36.25 
63.75 

33.75 
66.25 

47.75 
52.25 

47.25 
52.75 

48.0 
52.0 

52.25 
47.75 

57.5 
42.5 

614 

886 

1332 

39.5 
60.5 

49.75 
50.25 

43.0 
57.0 

55.75 
44.25 

02.75 
37.25 

62.5 
37.5 

64.5 
35.5 

66.75 
33.25 

718 

782 

1668 

A  STUDY   OF   THE   RESULTS. 

NUMBER. 

§  1.  The  two  upper  blocks  of  figures,  of  Tables  1  to 
7,  relate  to  the  "number  judgments"  obtained  in  Exper- 
iment A.  Examination  of  these  figures  discloses  that 
their  distribution  in  every  block  is  governed  by  certain 
main  laws,  all  holding  good  throughout,  thougli  with 
variable  force  in  their  relative  manifestations  under  the 
different  conditions  of  the  experiments.  We  have  to 
note  these  laws  and  to  inquire  their  meaning. 

The  first  is  a  law  of  chance,  imposed  by  tlie  methods 
of  the  experiment.  We  note  that  in  the  second  block 
of  figures,  through  all  the  tables,  the  values  in  the 
"  II  pin  "  or  left-hand  columns  are  invariably  plus,  and 
those  in  the  right-hand  or  "V  pin"  column  are  always 
minus.  The  reason  for  this  is  obvious  when  I  explain, 
that  the  subject  always  knew  what  categories  were 
being  used  in  the  experiments.^  In  the  present  "num- 
ber judgments  "  he  knew  there  could  never  l)e  less  than 
two  pins,  nor  more  than  five.     Consequently  there  could 

1  It  is  best  to  explain  these  to  the  subject  from  the  first,  as  it  is 
impossible  to  keep  him  from  forming  notions  about  them  diu-ing 
the  course  of  the  experiments. 


72  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

never  be  any  minus  errors  in  the  number  judgments  of 
"II  pins,"  and  never  any  plus  errors  in  those  of  '*V 
pins."  By  mathematical  calculation  we  could  theo- 
retically deduct  the  effects  of  chance  from  our  work ; 
but  as  we  should  then  have  a  set  of  figures  but  little  if 
any  more  significant  for  our  purpose  than  those  already 
given,  I  have  neglected  such  theoretical  calculations.^ 

Other  things  being  equal,  the  less  the  average  amount 
of  error  made  in  judging  the  pins,  the  greater  number 
of  times  per  hundred  should  the  pins  be  judged  cor- 
rectly. Consequently  the  distribution  of  the  figures 
in  the  upper  row  of  blocks  ought  ahvays  to  stand  in 
inverse  ratio  to  that  of  the  corresponding  figures  of 
the  blocks  below.  Examination  will  show  this  to  be 
the  case. 

§  2.  If  we  again  study  the  two  upper  blocks  in  our 
first  seven  tables,  we  shall  note  the  second  of  our  three 
laws  of  distribution.  It  may  be  stated  as  follows  :,  The 
longer  the  distance,  the  more  accurate  the  judgments. 

1  A  few  points,  however,  may  well  be  borne  in  mind.  Namely  : 
that  in  the  second  row  of  blocks,  the  effect  of  chance  is  to  make 
the  values  in  the  "II  pin"  and  in  the  "III  pin"  columns,  respect- 
ively, as  much  too  great  (+)  as  in  the  "IV  pin"  and  "Vpin" 
columns  they  are  too  small  (— )  ;  and  also  that,  respectively,  the 
-f-  and  —  errors  of  III  and  IV  ought  to  be  less  than  the  like  errors 
of  II  and  V.  If  all  the  errors  were  solely  due  to  chance,  then  all 
the  values  of  II  and  III  should  be  -|- ,  and  all  of  IV  and  V  should 
be  — ,  and  II  and  III  should,  respectively,  balance  IV  and  V.  All 
deviations  from  such  distribution  must  indicate  the  influence  of 
other  laws  yet  to  be  determined. 


OUn    NOTIONS    OF    NUMBER    AND    SPACE.  i  6 

This  law  appears  simple  enough  so  long  as  we 
study  alone  the  judgments  made  of  II  pins,  and  with- 
out asking  whi/  any  such  law  ought  to  hold  good. 
Practically,  throughout  all  the  blocks  and  tables, 
the  '*II  pin"  columns  show  regularly  increasing  accu- 
racy as  the  distance  increases  from  1  to  3  cm.;  that 
is,  downward  in  these  columns.  Moreover,  the  "III 
pin "  columns  show  openly,  in  general,  a  tendency  to 
follow  this  same  law.  We  may  note,  however,  that 
usually,  throughout  the  shorter  distances,  these  "III 
pin "  judgments  incline  to  follow  an  opposite  course  ; 
beginning  at  the  top  of  these  columns  the  accuracy  of 
judgment  appears  to  decrease,  till  a  certain  length  of 
distance  is  reached  (differing  according  to  the  region  of 
body  studied),  whence  onward,  with  increasing  distance, 
the  figures  follow  the  law  at  present  in  hand  as  regu- 
larly as  do  the  judgments  of  II  pins  throughout.  Al- 
ready this  regularity  of  exception  to  our  present  law 
foreshadows  the  cooperation  of  a  third  law.  But  we 
feel  much  more  the  need  of  some  such  further  principle 
to  account  for  our  results,  as  we  come  to  examine 
columns  IV  and  V.  Here  we  appear  to  have,  com- 
pletely reversed,  the  very  common  laAv  of  experience, 
which  certainly  holds  good  for  two  pins,  and  if  we 
accepted  the  mere  empirical  evidence  of  these  figures, 
we  should  conclude  that,  in  judging  four  and  five  piris, 
the  effect  of  spreading  them  further  apart  decreased 


74  OUE    NOTIONS    OF    NUMBER    AND    SPACE. 

the  accuracy  of  our  judgments.  This  brings  us  to  our 
Third  Law. 

§  3.  If  we  say  that,  with  decreasing  distance,  uncer- 
tainty of  judgment  increases,  we  shall  merely  be  stating 
our  old  Law  Two  in  a  new  form.  But  ''increasing 
uncertainty"  means  increasing  liability  to  spread  our 
judgments  over  categories  other  than  the  correct  one  ; 
it  means  increasing  tendency  for  the  mind  to  pitch  its 
judgments  upon  its  possible  categories,  rather  than 
upon  the  right  particular  one ;  to  depart  from  highly 
specialized,  accurate,  and  fixed  habit,  to  less  definite, 
and  less  rigidly  developed  habit. 

If  now,  according  to  Law  Two,  with  decreasing  dis- 
tance we  have  inci'easing  uncertainty,  then,  by  Law 
Three,  with  decreasing  distance  we  have  a  tendency  to 
spread  the  increasing  uncertainty  more  and  more  over 
the  wider  field  of  possible  judgments. 

By  ''  possible "  Ave  must  not  mean,  as  limited  to  the 
four  number  categories  of  pins  in  our  experiment,  but 
possible  by  our  whole  nature.  The  whole  range  of 
numerical  categories,  which  is  possible  in  this  larger 
sense,  is  very  great,  and  the  relaxation  of  accuracy  or 
of  particular  habits  does  not  take  place  in  equal  pro- 
portion toward  all.  For  the  four  number  categories 
used  in  our  experiments,  the  uncertain  judgments  con- 
tinually drift  toward  higher  numerical  categories.  The 
lower  the  numerical  category  the  stronger  is  this  tend- 


OUK    NOTIONS    OF    NUMBER    AND    SPACE.  75 

ency.  Or  at  least,  the  drift  of  errors  being  in  a 
general  direction,  we  find  as  the  outer  impression  is 
moved  in  that  direction,  the  less  becomes  the  error 
which  is  due  to  the  drift  of  uncertainty.  It  is  probable 
that  if  the  numerical  categories  of  our  experiment 
extended  high  enough,  we  should  find  a  place  where 
there  Avould  be  no  tendency  for  uncertain  judgments 
to  drift  in  any  single  direction.  The  reason  for  all 
this  we  shall  come  to  presently,  but  we  may  now  throw 
our  Third  Law  into  its  empirical  form  as  follows : 
The  lower  the  numerical  category,  the  stronger  is  the 
tendency  of  the  uncertain  judgments  to  drift  toward 
overestimation. 

§  4.  Having  found  our  three  laws  we  will  now  ex- 
amine our  tables.  We  shall  need,  here,  to  follow  in 
detail  but  a  single  example,  and  simply  because  they 
are  longer  than  some  of  the  others,  we  will  take  Blocks 
65  and  70  in  Table  4.  Beginning  in  the  upper  left- 
hand  corner  of  65  we  find  that,  for  the  region  here 
studied,  two  pins,  when  separated  by  a  distance  of 
1  cm.,  are  judged  correctly  only  seven  times  out  of  a 
hundred.  From  the  figure  7  vertically  downward,  the 
numbers  increase  pretty  regularly  till  at  5  cm.  two 
pins  are  judged  correctly  seventy-nine  times.  All  this 
plainly,  by  reason  of  Law  Two.  Going  back  to  the 
same  figure  7,  we  see  the  numbers  increasing  hori- 
zontally to  the   right,  till,  with   the   shortest  distance 


76  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

category  of  1  cm.  remaining  constant,  we  find  V  pins 
judged  correctly  fifty -five  times.  All  this,  however,  as 
I  believe,  wholly  from  the  ''drift  of  error,"  and  not  as 
in  column  II  through  increased  functional  accuracy  of 
judgment.  Merely,  here,  Law  Three  overbalances  Law 
Two,  As  we  run  down  the  left-hand  column  V,  we  see 
the  number  decrease  pretty  regularly  from  55  to  28. 
This  indicates  no  contradiction  or  suspension  of  Law 
Two,  but  by  Law  Three,  with  increasing  distance  should 
go  decreasing  drift  of  error;  that  is,  less  tendency  of 
the  uncertain  judgments  toward  overestimation.  If  we 
look  below,  in  Block  70,  at  the  corresponding  figures  to 
the  last  above-mentioned,  it  would  seem  at  first  sight 
absurdly  incorrect  to  assert  overestimation  for  these 
judgments  at  all,  for  all  the  amounts  of  error  are  given 
as  minus.  But  plainly  this  is  explained  by  Law  One. 
By  the  terms  of  the  experiment  there  could  be  no 
judgment  greater  than  five  pins.  But  the  "drift  of 
error,"  by  Law  Three,  was  active  all  the  time,  even  as 
against  these  terms,  and  made  the  values  appearing 
here  as  relatively  minus,  really  greater  ;  that  is,  less 
minus  than  they  otherwise  would  have  been.  As  evi- 
dence that  the  natural  drift  of  error  for  the  whole 
range  of  numerical  judgments  from  II  to  V  inclusive  is 
positively  upward,  it  is  to  be  noted  that  the  average 
error  for  all  the  distances,  if  calculated,  would  be  plus ; 
that  the  average  for  all  distances  is  + 11.44 ;  and  that 


OUR    NOTIONS    OF    NUMBER    AND    SPACK.  77 

a  roughly  estimated  (leduction  of  the  "chance"  values 
from  the  various  coiumns,  as  suggested  in  our  discussion 
of  Law  One,  would  leave  an  unmistakable  indication 
that  the  real  tendency  in  this  range  is  throughout 
toward  overestimation. 

The  blocks  above  examined  are  typical,  and  Ave 
may  observe  of  the  number  judgments  throughoiit 
our  tables,  that  in  column  II  the  obviously  controlling 
influence  is  Law  Two  :  with  increasing  distance  goes 
increasing  accuracy.  In  column  V  the  obvious  influence 
is  Law  Three :  with  increasing  distance  goes  decreas- 
ing uncertainty ;  therefore  decreasing  drift  toward  over- 
estimation;  therefore  decreasing  correction  of  functional 
inaccuracy  by  "local  drift";  therefore  fewer  correct  judg- 
ments—  the  influence  of  Law  Two  outweighing  that  of 
Law  Three,  the  latter  remaining  active  all  the  while. 
In  column  IV,  as  it  should  be.  Law  Three  is  less 
powerful  than  in  V,  but  remains  the  obvious  influence. 
In  column  III,  as  it  should  be.  Law  Three  is  strongest 
in  the  shorter  distances,  and  with  decreasing  force  is 
dominant  up  to  a  distance  (about  2.5  cm.  in  the  figures 
for  the  abdomen)  when  its  influence  gives  way  to  that 
of  Law  Two,  which  holds  sway  thence  upward.  The 
influence  of  Law  One  we  have  already  made  sufficiently 
obvious. 

§  5.  Having  enabled  the  reader  to  study  our  tables 
for  himself,  I   shall  now  dare  to  ask  him  to  consider 


78       oun  NOTIONS  of  number  and  space. 

their  content  and  their  laws  in  the  light  of  a  funda- 
mental hypothesis  as  to  the  fornitition  of  numerical 
judgments  in  general.^ 

Attacking  our  Law  Tavo,  to  discover  why  with  in- 
creasing distance  there  should  go  increasing  accuracy 
of  judgment,  we  must  first  ask  why  the  simultaneous 
stimulation  of  two  points  of  skin  lying  under  two  pin- 
points, situate  a  proper  distance  apart,  should  ever  give 
rise  to  the  conception  of  duality  at  all. 

The  reason  for  this  is  somewhat  as  follows.  In  the 
first  place  these  two  areas  must  previoush^,  sometime 
in  life,  have  been  stimulated  the  one  after  the  other  in 
immediate  succession.  The  conception  of  duality  in 
general  is  a  particular  mental  state  or  process  which  is 
the  result  of  such  a  shock  ;  it  is  the  feeling  of  such  a 
shock.  As  such  it  falls  under  all  the  laws  of  Associa- 
tion and  of  Memory  which  govern  other  conceptions, 
notions,  mental  states,  and  processes.  As  such,  regain, 
it  is  subject  to  all  the  eifects  of  habit  which,  going 
back  a  step  further,  govern  these  above  laws  ;  and  in 
turn  the  particular  habits,  which  control  the  associative 
and  perceptive  activities  of  any  conception  of  duality 
for  any  particular  pair  of  points  or  areas  in  an}^ 
given   region,  depend    still    more    fundamentally   upon 

1  I  find  the  main  thoughts  of  this  hypothesis  best  stated  in 
Professor  James'  Principles  of  Psychology  (vol.  I,  chap,  xi,  in 
particular,  pages  487,  488,  498). 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.  79 

the  average  run  of  experience  common  between  these 
two  points  or  areas.  The  final  habit  is  the  resultant 
average  of  all  the  past  liabits.  On  the  whole,  during 
life  these  particular  areas  have  not  been  as  frequently 
stimulated  simultaneously  as  successively;  consequently 
the  successive  '-'number  mode"  which  is  the  conception 
of  duality,  has  become  more  strongly  established  as 
between  these  two  points,  than  has  the  mode  native 
to  simultaneous  stimulations,  namely,  the  conception  of 
unity.  We  could,  perhaps,  by  refined  means,  stimulate 
the  total  nerve  ends  of  even  these  two  pin-point  areas 
in  some  plural  form  of  succession.  That  is,  we  could 
divide  them  into  three,  four,  or  any  number  of  separate 
groups  and  then  stimulate  these  groups  in  succession. 
And  if  such  a  practice  prevailed  through  life  above  all 
other  modes  of  stimulating  these  areas  collectively, 
then  by  our  hypothesis,  the  simultaneous  pressure  of 
two  pins  upon  these  separate  areas  would  give  us  the 
numerical  conception  corresponding  to  that  mode  of 
succession,  rather  than  as  now  to  the  dual  mode.  AVhy, 
therefore,  the  pressure  of  two  pins  on  separate  areas 
commonly  give  us  a  conception  of  their  duality,  is 
not  so  much  that  the  particular  tools  of  stimulation 
then  and  there  used  are  two  pins,  as  that  between  those 
precise  areas  taken  collectively  the  mode  of  stimulation, 
which,  on  the  whole,  through  life  has  prevailed  and  set 
up  its  particular  habit  of  mental  reaction,  has  been  the 
mode  of  dual  succession  above  all  others. 


80  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

This  being  so,  the  explanation  of  our  Law  Two  (that 
with  increasing  distance  the  habits  of  plural  numerical 
judgments  become  more  accurate)  is  easily  reached.  In 
very  small  areas  the  nerve  ends  collectively  are  more 
frequently  stimulated  together  than  separately.  The 
habit  of  unity  prevails  over  all  other  numerical  habits. 
Hence  stimulation  of  such  areas  is  most  likely  to  give 
rise  to  the  numerical  conception  of  "one  thing."  This 
will  hold  good  even  though  the  tools  of  stimulation  be 
the  same  two  pins  which,  when  set  further  apart,  will 
invariably  give  rise  to  the  notion  of  their  being  two ;  if 
the  two  pins  be  set  too  near  together  they  will  ordi- 
narily be  judged  as  "one."  When,  now,  we  come  to 
spread  this  spacing  toward  wider  distances,  Ave  depart 
from  conditions  where  the  unitary  habit  is  strong 
toward  those  where  this  habit  is  less  strong,  and  where 
the  habit  of  duality  begins  to  be  its  rival.  As  we  go  on 
widening  we  find  the  former  continually  weakening  till 
it  fails  entirely,  and  the  latter  growing  more  and  more 
strong  till  its  judgments  approximate  absolute  certainty 
and  accuracy. 

Our  Law  Two,  therefore,  in  so  far  as  it  relates  to 
our  judgments  of  two  pins,  is  but  an  expression  of 
the  fundamental  fact  that  the  further  two  points  are 
separated  on  the  skin,  the  more  confirmed  become  the 
resultant  habits  of  experience,  relative  to  those  points, 
in  favor  of  the  dual  mode  of  reaction  over  and  above  all 
other  modes  of  numerical  reaction. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.  81 

The  law  holds  equally  good  and  expresses  precisely 
similar  facts  in  higher  numerical  judgments.  Judg- 
ments of  "three,"  and  of  ''four/'  are  particular  mental 
states  or  conceptions,  based  upon  series  of  three,  and  of 
four  cuccessive  sense  impressions,  in  a  manner  strictly 
analogous  to  that  in  which  judgments  of  "two"  are 
formed.  Other  things  being  equal,  the  further  apart 
any  x  number  of  separate  points  or  areas  of  skin  shall 
be,  the  more  through  life  are  all  those  stimulations  or 
impressions,  which  bring  all  those  points  into  collective 
relationship  with  one  another,  likely  to  fall  into  a  series 
of  X  successive  impressions  rather  than  into  any  other 
particular  combination  collectively  of  those  several 
points.  The  influence  of  this  truth  upon  our  mental 
habits  works  as  a  factor  in  the  formation  of  the  judg- 
ments of  one  numerical  category  as  certainly  as  in  the 
formation  of  those  of  another.  But  while  it  may  be  an 
unmixed  influence  in  one  category — such  as  we  discov- 
ered it  to  be  in  the  "II  Pin"  columns  of  our  tables — it 
may  be  mixed  with  other  influences  in  judgments  of 
other  categories,  such  as,  for  instance,  those  of  columns 
IV  and  Y.  It  remains  to  glance  at  some  of  these  other 
influences,  in  the  light  of  our  general  hypothesis. 

§  6.  One  of  these  is  the  influence,  under  Law  Three, 
by  reason  of  which  the  drift  of  errors  in  uncertain 
judgments  (at  least  in  those  for  the  categories  used  in 
our  experiments)  is  constantly  toward  overestimation. 


82  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

We  have  pointed  out  that  in  departing  from  liabitual 
reaction  in  a  single  way,  we  sink  to  a  looser  and  wider 
range  of  reactions.  If  the  bond  of  connection  or  of 
associative  habit  for  the  uncertain  sense  impressions 
were  equally  strong  toward  all  the  possible  numerical 
reactions,  then  it  would  be  easy  to  see  why,  for  judg- 
ments of  the  lower  numbers,  the  average  drift  of  error 
would  constantly  be  toward  over-estimation.  It  would 
be  a  mere  matter  of  mathematical  chance,  or,  as  we 
might  say  here,  of  psychological  chance.  The  drift 
being  equal,  the  average  drift  for  the  lower  numbers 
would  necessarily  be  upward.  How  far  this  serves  to 
explain  the  actual  drift  of  the  errors  of  uncertainty  in 
our  experiments,  or  whether  it  is  a  correct  explanation 
at  all,  I  cannot  at  present  with  any  certainty  decide. 
Personally,  however,  I  incline  to  look  upon  this  drift  as 
the  expression  of  a  loose  and  inexact  general  habit  of 
the  whole  brain,  or  of  a  large  sphere  of  it,  to  act  most 
strongly  in  the  direction  of  the  general  average  of  the 
entire  range  of  experiences  embraced  in  that  general 
habit ;  this  rather  than  to  conceive  of  the  many  errors, 
actually  committed  in  uncertainty,  as  so  many  accidental 
reactions  in  several  loosely  incitable  but  definitely 
directed  habits. 

§  7.  Further  evidence  for  our  hypothesis  appears 
when  Ave  compare  the  results  obtained  upon  one  region 
of  the  body  with  those  of  another,  but  for  reasons  that 


OUR    NOTIONS    OF    NtJ^NIBER    AND    SPACE.  83 

will  become   evident  I  will  postpone  considering  this 
matter. 

§  8.  With  reference  to  Tables  5  and  6,  both  refer  to 
the  same  region  of  skin  —  the  forearm.  The  difference 
between  the  method  pursued  in  the  regular  experiments 
upon  the  forearm  (Table  3)  and  that  which  gave  us 
Table  o  is,  that  in  the  former  the ,  row  of  pins  was 
rocked  lengthwise  upon  the  skin,  and  in  the  latter  all 
the  pins  were  pressed  on  evenly  and  at  once.  The 
former  gave  series  of  impressions  precisely  like  those 
which  originally  gave  rise  to  our  numerical  judgments. 
They  are  such  original  impressions  as  our  habits  of 
numerical  judgment  at  first  hand  are  founded  upon. 
The  latter  method  gave  us  only  simultaneous  impres- 
sions ;  these  were  no  longer  impressions  like  the 
original  impressions ;  were  not  the  old  successions 
happening  over  again  ;  and  the  judgments  Avhich 
followed  were  only  weakened  imitations  of  former 
judgments  awakened  at  second-hand  through  memory 
and  association.  Xow,  since  these  latter  judgments 
depend  more  upon  memory  than  do  the  former,  they 
must  be  more  uncertain  tlian  judgments  based  upon 
successive  impressions.  But  with  increased  uncertainty- 
should  go  more  marked  exhibition  of  Law  Three  ;  and  if 
in  our  tables  we  discover  this,  we  should  count  it  as 
confirmatory  of  our  general  hypothesis  as  outlined  from 
the  beginning  of  our  paper.  '■-' 


\ 


84  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

Turning  to  Block  83  of  Table  5  we  do  observe  just 
such  an  increased  influence  of  Law  Three  in  proportion 
to  the  relative  influence  of  Law  Two,  as  we  have  spoken 
of.  Comparing  these  figures  Avith  the  corresponding 
ones  for  our  "  regular "  experiment  on  the  forearm 
(Block  45,  Table  3),  we  find  unmistakable  evidence  of 
greater  uncertainty  and  of  the  distribution  of  the  conse- 
quent errors  according  to  the  law  which  we  have  laid 
down  and  given  the  reasons  for.  In  column  II,  where 
the  effect  of  Law  Two  always  is  most  obvious,  we  see 
the  accuracy  of  judgment  reduced  by  some  30  to  50  per 
cent.,  while  in  column  V,  where  Law  Three  is  most 
evident,  we  find  an  increase  of  its  influence  about  the 
same  in  amount.  Moreover,  as  the  method  of  applying 
the  pins  was  changed  alike  for  the  whole  scale  of 
distances,  so  the  results  show  a  tolerably  equal  amount 
of  change  throughout  the  block ;  that  is,  the  increased 
drift  toward  over -estimation  is  a  pretty  equal  one 
throughout.  All  the  numbers  in  column  V  are  approxi- 
mately as  much  increased  as  those  in  column  II  are 
decreased.  In  short,  throughout,  with  a  like  change 
of  method  we  see  a  like  change  due  to  Law  Three. 

§  9.  For  the  results  shown  in  Table  6,  not  only  were 
the  pins  applied  evenly  without  rocking  but,  by  the 
effort  of  will,  the  attention  was  confined  strictly  to  the 
number  of  pin-points  clearly  felt.  Practically,  this 
means  that  the  mind  was  not  permitted  to  range  up  and 


OUR   NOTIONS    OF    NU:*LBER    AND    SPACE.  85 

down  the  line  of  pins,  as  it  sat  on  the  skin,  "listening" 
here  and  there  as  to  whether  a  pin  really  was  felt  then 
or  not,  and,  by  a  continuation  of  this  process,  coupled 
with  a  knowledge  which  the  subject  always  had  of  all 
the  categories  which  were  being  used,  to  reckon  out  just 
what  combination  of  pins  and  distance  was  at  the 
moment  being  applied.  I  say  the  mind  was  not  per- 
mitted to  range  up  and  down.  This  is  but  saying  that 
the  fundamental  process  Avhich  is  the  basis  of  Law 
Three  was  here  not  permitted  to  play.  The  memory 
process  was  shut  out,  and  consequently  the  drift  toward 
over-estimation  was  shut  out. 

Xot  only  this,  but  I  am  inclined  to  think  that  the 
actual  results  reveal  to  us,  in  a  strikingly  significant 
manner,  the  working  of  memory  in  reviving  through 
simultaneous  impressions  the  numerical  judgments  which 
originally,  at  least,  were  the  effects  of  successive 
impressions.  The  "  ranging  up  and  down  of  the  mind  " 
in  search  of  the  proper  category  is  much  like  actually 
playing  over  again  in  imaginary  processes,  the  actual 
successions  of  the  original  events  ;  and  the  effects  of 
inhibiting  this  '■  ranging,"  as  shown  in  Table  6,  are  so 
surprising  as  to  emphasize  the  question  as  to  whether 
or  not  such  imaginary  successions,  in  some  form,  per- 
haps almost  infinitely  compressed,  are  not  what  really 
happen,  in  the  formation  of  all  numerical  judgment 
from  a  simultaneous  impression,  and  whether,  therefore, 


86  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

they  are  not  absolutely  necessary  to  the  formation  of 
such  judgments  ?  Under  the  influence  of  the  inhibition 
of  such  processes,  column  V  of  Table  G  shows,  in  some 
places,  an  absolute  lack  of  correct  judgments  of  the 
higher  categories  —  those  which  would  require  the 
greatest  amount  of  this  imaginary  play  —  and  shows 
but  a  very  small  number  of  such  judgments  anywhere 
throughout  the  column.  iVTothing  I  have  said,  however, 
must  be  mistaken  for  a  premature  inclination  to  decide 
this  matter.^ 

DISTANCE. 

§  10.  When  we  draw  a  pencil-point  along  the  skin 
we  stimulate  successively  the  nerve  ends  lying  in  the 
line  drawn.  In  essential  nature  such  an  event  is  no 
different  from  those  primitive  occurrences  which,  accord- 
ing to  our  hypothesis,  give  origin  to  our  number  judg- 
ments. They  are  both  based  on  serial  impressions. 
The  difference  between  the  number  judgments  and  the 
distance  judgments  lies  chiefly  in  the  nature  of  the 
successions  which  characterize  each.  In  the  former 
the  terms  of  the  series  are  comparatively  few,  the  suc- 
cessions are  sharply  marked-off,  and  slow  ;  so  slow  and 
marked  that  we  note  and  count  them  —  successively  give 

1 1  am  sure  that  our  experiments  here  are  capable  of  teaching  us 
an  important  lesson  as  to  the  intimate  nature  of  the  processes  of 
attention  in  the  formation  of  judgments  in  general. 


OUK    ]S'OT10^'S    OF    ^-U:MBER    AND    SPACE.  87 

names  to  them.  Thus  :  one  —  two  ;  or  one  —  two  — 
three.  When  a  line  is  drawn  along  the  skin  a  myriad 
of  nerve  ends  are  stimulated  in  relatively  rapid  and 
unmarked  successions  ;  so  rapid  and  unbroken  that 
we  do  not  note  the  separate  terms,  nor  individually 
count  them.  The  difference  is  that  in  one  case  we 
say,  "  one  —  two  —  three,"  and  in  tlie  other,  ''  so 
ma-a-a-a-any  ''  or  '•  so  fa-a-a-a-a-a-a-ar." 

Each  specific  length,  however,  of  the  distance  series 
has  a  nature  of  its  own  which  is  the  basis  of  each  one 
of  our  specific  categories  of  distance  judgment  (our 
ideas  of  particular  distance),  in  the  same  Avay  tliat  the 
number  series  each  have  a  particular  length  or  number 
of  terms  which  is  the  basis  of  each  category  of  number 
judgments. 

§  11.  We  have  said  that,  given  a  definite  set  or 
arrangement  of  nerve  ends,  until  this  set,  some  time  in 
life,  be  broken  up  and  its  parts  be  first  stimulated  in 
some  sort  of  succession,  any  sort  of  simultaneous  stimu- 
lation of  this  particular  set  will  not  arouse  any  sort  of 
plural  category  whatever.  AVe  now  say  that  it  will 
arouse  no  conception  of  distance  whatever.  Any  set  of . 
nerve  ends  which  has  never  been  stimulated  except  in 
complete  simultaneity  wall  give  us  the  same  sort  of 
mental  response  or  experience  when  distributed  over 
the  surface  of  the  skin  in  a  compact  bunch,  as  when 
distributed  in  anv  sort  of  lineal  arrangement. 


88  (3UR    NOTIONS    OF    NUMBER    AND    SPACE. 

§  12.  Yet  it  is  the  fixed  lineal  or  spatial  arrange- 
ment of  our  dermal  nerve  ends  that  determines  our 
particular  distance  and  space  conceptions  regarding 
them.  This  happens  because  it  is  the  fixedness  of  each 
particular  arrangement  that  determines  what  manner  of 
serial  stimulation  through  life  shall  most  frequently  fall 
to  the  lot  of  that  collective  group  of  nerve  ends.  If 
two  nerve  ends  are  permanently  located  immediately 
beside  each  other  they  are  likely  to  be  stimulated  more 
times  during  life  simnltaneonsly  than  successively ;  and 
consequently,  upon  simultaneous  stimulation,  are  more 
likely  not  to  arouse  any  notion  of  distance  betv/een 
these  two  points  than  to  arouse  such.  If  two  nerves  are 
fixed  widely  apart  they  are  more  likely  than  otherwise, 
in  the  whole  of  life,  to  be  affected  by  various  moving 
impressions  which  shall  first  stimiilate  one  nerve  at  a 
given  point  in  the  series  of  impressions  and  the  other 
nerve  at  another  point  in  the  series.  Consequently  it 
is  more  likely  than  not  that  some  sort  of  distance  cate- 
gory will  be  developed  and  attached  to  the  collective 
stimulation  of  these  two  separate  points. 

§  13.  Every  pair  of  separate  points  is  likely  to  be 
stimulated  by  all  sorts  of  lineal  impressions  moving 
first  through  one  point  and  then  through  the  other. 
This,  in  the  same  way  that  it  is  possible  to  draw  all 
sorts  of  lines  (straight,  broken,  and  curved)  through  any 
two  points  of  our  skin.     The  question  then  arises  how 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.  89 

any  particular  distance  category,  corresponding  to  some 
particular  length  of  moving  series,  comes  to  be  so  joined 
to  any  two  particular  points  that  we  commonly  judge 
them  to  be  a  definite,  actual  distance  apart? 

This  is  not  difficult  to  answer  when  we  recall  that 
the  kind  of  memory  category  that  is  awakened  by  the 
simultaneous  stimulation  of  any  definite  combination  of 
nerve  ends,  is  based  on  the  dominant  and  average  habit 
which  is  the  resultant  of  the  experiences  which  have 
most  frequently  combined  that  particular  collection  of 
nerve  ends.  With  reference  to  the  skin,  it  is  evident 
that  right-line  movements  are  likely  to  prevail  between 
points  separated  by  a  few  centimeters  (as  in  our  experi- 
ments) far  and  away  above  any  other  particular  form  of 
lineal  movement. 

§  14.  But  the  rate  of  drawing  a  line  may  be  infi- 
nitely variable,  and  the  time  element  of  the  series  is  the 
most  important  thing  of  all,  according  to  our  hypothesis, 
as  the  primitive  basis  of  distance  measurement.  Here, 
again,  we  see  that  the  single  definite  habit,  which  finally 
results  froin  the  particular  modifications  of  each  one  of 
the  infinite  number  of  infinitely  varied  time  series  expe- 
rienced through  life  between  every  pair  of  dermal  points, 
solves  the  difficulty.  The  average  of  such  an  infinity  of 
time  experiences  for  any  pair  of  points  would,  other 
things  being  equal,  be  proportional  to  the  actual,  fixed 
right-line  distance  between  the  points.     Consequently, 


90  OUE,    NOTIONS    OF    NUMBER    AND    SPACE. 

iu  proportion  as  the  resultant  or  prevailing  liabit  is  a 
fixed  and  accurate  one  does  its  mental  judgment  or 
perception  accurately  represent  or  correspond  to  the 
actual  right-line  distance. 

§  l.j.  At  this  point  many  difficulties  arise  if  ^ve 
inquire  as  to  the  intimate  and  specific  nature  of  our 
different  distance  perceptions.  All  that  we  have  said 
supposes  them  to  be  based  upon  reawakened  time  series 
of  correspondingly  specific  length  or  nature.  Yet  we 
surely  must  reject  the  notion  that  our  hasty  judgments 
of  different  distances  are  always  of  the  same  absolute 
and  specific  time  lengths,  all  in  due  proportions.  But 
if  not,  how  do  they  preserve  any  sort  of  proportions 
between  themselves,  duly  representative  of  actual  outer 
differences  ?  This  leads  us  to  one  of  the  most  obscure 
regions  of  psychology. 

To  me  the  following  hypothesis  seems  both  more 
clear  and  more  justifiable  than  the  average  psycho- 
logical hypothesis  of  modern  text  books  relative  to  the 
intrinsic  nature  of  a  judgment.  It  is  the  connective 
or  associative  function  of  any  mental  processes  or  habit 
that  is  of  importance  in  the  formation  of  accurate 
thoughts  and  judgments,  rather  than  the  nature  of  its 
content.  If  the  function  is  accurate  and  specific,  the 
judgment  will  be  accurate  and  specific.  The  function  of 
the  specific  perception  or  judgment  is,  to  make  the 
proper  connection  between  the  outer  event  or  impres- 


OUR    NOTIONS    OF    NUMBER    AND    STACK.  01 

sion  and  certain  following  thoughts,  perceptions,  or 
associations  about  that  event  or  impression.  If  the 
proper  specilic  connection  is  made,  it  makes  no,  or  little, 
difference  to  the  accuracy  of  the  thinking,  what  the 
specific  nature  of  the  mental  content  of  the  judgmental 
in  itself  may  be,  either  qualitatively  or  in  absolute  time- 
duration.  Suppose  the  perceptive  connection  is  to  be 
with  the  motor  idea  which  incites  us  to  say,  "  one 
centimeter.'"'  The  connective  activity  may  occupy  an 
absolutely  longer  or  shorter  time,  and  this  make  no 
difference  provided  the  nature  of  the  process  is  such 
that  the  proper  motor-idea  is  eventually  incited.  This 
being  so,  I  think  we  may  easily  conceive  how  that, 
though  our  various  specific  judgments  are  all  based  upon 
habits  which  are  the  resultants  and  the  correspondents 
of  time  series  of  relatively  different  absolute  lengths, 
they  yet  may  not  themselves  occupy  absolute  intervals 
all  proportionally  different,  nor  their  conscious  content 
be  of  any  given  specific  phenomenal  nature.  Indeed  we 
may  easily  conceive  how  the  really  decisive  and  dis- 
tinctive link  should  be  wholly  an  unconscious  mental 
process.  If  we  take  our  stand  on  the  Summation 
Theory,  we  can  conceive  how  the  specific  intensive 
force  or  constituency  of  the  differently  summated  series 
might  determine  the  proper  connection  independently 
of  specific  time  dvirations.  Or  we  may  with  plausibility 
conceive  of   some    specific  anatomical    arrangement    of 


92  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

brain  parts,  correspondent  to  each  specific  judgment, 
whose  liability  to  be  affected  as  a  whole,  depended  upon 
specific  series  of  stimulations  of  absolute  length,  and 
yet  whose  subsequent  activity  as  specific  wholes  in 
memory  should  not  necessarily  occupy  always  the  same 
absolute  time-rhythms.  Or,  better  still,  perhaps  we  may 
conceive  of  a  combination  of  intensive,  anatomical,  and 
spatial  distributions  which  shall  mediate  the  activities 
correspondent  to  the  correct  specific  judgments  and 
make  accurate  connection  thereby  with  the  proper 
associative  thoughts,  yet  do  this  quite  independently  of 
an  absolute  time  performance  which  should  be  pre- 
served constant  for  every  repetition  of  the  judgment. 

§  16.  With  so  much  of  our  hypothesis  before  us  we 
may  now  come  nearer  the  laws  which  specially  govern 
our  tables.  To  begin  with,  we  may  note  that  in  the 
main  all  we  have  said  about  ''chance"  must  hold  good 
as  much  for  the  two  lower  rows  of  blocks  in  our  tables 
— those  which  relate  to  the  distance  judgments  —  as  it 
does  for  the  two  upper  rows,  which  relate  to  the  number 
judgments. 

§  17.  According  to  Law  Two  the  numerical  judg- 
ments were  the  more  accurate,  the  greater  was  the 
distance.  Under  what  we  may  call  the  same  law, 
we  sliall  now  find  the  accuracy  of  the  distance  judg- 
ments also,  as  a  rule,  increasing  with  tlie  actual 
distance.      But   the   reasons  for  this   law  are   neither 


OUR    NOTIONS    OF    NUMBEU    AND    SPACE.  93 

SO  plain  nor  so  simple  as  previously.  Since  our  percep- 
tion of  the  distance  between  two  points  of  skin  is  based, 
primitively,  upon  the  predominance  of  movements 
in  the  right-line  joining  those  points  over  moverdents 
in  any  other  particular  line  or  combination  of  lines  ; 
and  since  the  greater  the  distance  the  less  should 
be  this  predominance,  therefore,  were  these  the  only 
principles  determining  Law  Two  for  distance,  we  ought 
to  find  increasing  inaccuracy  with  increasing  distance. 
It  is  probable  that  this  principle  has  its  proper  influ- 
ence, primitively,  in  developing  our  notions  of  particular 
distances  everywhere,  and  very  likely  there  are  lengths 
of  distance  for  certain  stretches  or  regions  of  skin 
where  the,  principle  is  a  predominating  one;  but 
within  the  short  distances  and  regions  investigated 
by  us,  other  principles  come  in  which  counteract  and 
far  outweigh  it. 

§  IS.  One  of  tliese  other  principles  is  similar  to  the 
one  which  explained  Law  Two  under  "  Number."  We 
may  describe  it  as  follows  :  Our  memories  concerning 
the  distance  between  two  points  are  based  on  move- 
ments between  these  points ;  the  longer  the  distance,  the 
more  are  the  two  points  disassociated  by  these  move- 
ments; that  is,  the  more  do  these  movement-habits  weigh 
against  the  tendency  for  the  two  points  to  fuse  spatially 
in  memory ;  they  tend  so  to  fuse  in  proportion  as  the 
points  are  habitually  combined  simultaneously.     There- 


94  OUR    NOTIONS    OF    NUJNIBER    AND    SrACE. 

fore,  in  forming  the  prevailing  memory-habit,  tlie  disas- 
sociation  of  movement  is  always  opposed  to  the  influence 
of  simultaneous  combination.  Thus  Law  Two  in  dis- 
tance, or  at  least  within  short  distances,  formulates,  as  in 
number,  the  increasing  tendency  of  the  influences  of  the 
successive  experiences  of  life  to  prevail  w^th  increasing 
actual  distance  over  the  influence  of  the  simultaneous 
experiences  of  life.  Put  into  every-day  language,  this 
is  but  saying  that  Ave  are  more  likely  to  judge  the 
distance  between  two  points  accurately,  the  more  capa- 
ble we  are  of  forming  a  distinct  conception  of  their 
sej^arateness. 

Put  more  technically:  the  further  apart  two  points  of 
skin  are,  not  only  is  our  conception  of  their  ^eparateness 
heightened  through  disassociation  due  to  the  movements 
between  those  single  points,  but  also  it  is  heightened 
through  the  growing  differentiation  in  the  grouping  of 
the  local  associations  around  each  point.  If  the  di^stance 
increase  enough,  the  two  points  w^ill  habitually  fall  into 
strikingly  incongruous  groups.  Por  instance,  the  point 
of  the  toe  and  the  back  of  the  head.  Now,  it  is  plain 
to  any  one  having  much  experience  as  a  subject  in 
the  experiment  we  are  discussing,  that  our  judgments 
are  not  confined  to  data  based  on  the  precise  line  join- 
ing the  terminal  pins.  Rather,  we  form  a  notion  of  the 
separate  and  particular  region  w^here  we  find  one  pin  to 
be,  then  another  notion  of  the  region  of  the  other  pin, 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.  95 

and  we  then  say  that  one  region  is  ''  so  far  "  from  the 
other.  This  being  so,  we  see  at  once  that  the  particuhir 
judgment  of  distance  which  we  finally  render,  is  not 
based  on  the  simple  disassociation-force  of  the  single 
line  between  the  two  pins,  but  by  the  whole  disassocia- 
tion-force of  the  two  "region  groupings" — the  whole 
force  of  the  region's  associations  to  fall  into  distinct 
and  separate  groupings  rather  than  to  fuse  into  an 
unseparated  single  spot. 

§  19.  Still  another  principle  works  in  favor  of  Law 
Two.  Our  voluntary  measurements  of  shortest-distance 
are  always  in  straight  lines.  When  we  train  our  judg- 
ments of  distances,  we  are  always  training  our  right-line 
memories.  We  know  how  great  are  the  results  of  train- 
ing, and  how  great  must  be  the  influence  through  life  of 
joining  the  right-line  memory  to,  and  weaving  it  into, 
the  concept  of  measuring. 

§  20.  I  think  it  should  now  be  clear  how  increasing 
distance  works  to  increase  the  accuracy  of  estimating 
the  distance  between  points.  The  greater  the  distance, 
the  sharper  and  stronger  will  be  our  conception  of  the 
separateness  of  the  two  regions.  The  sharper  and 
stronger  this  conception  of  the  regions,  the  more  highly 
developed  in  connection  therewith  will  be  the  memory 
effects  of  the  voluntary  measuring  of  the  points.  The 
voluntary  measurements  will  all  be  based  on  right-line 
experiences.      The  three  concepts  —  of  the  regions,  of 


96  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

the  measuring,  and  of  the  right-line  —  will  fuse,  and 
act  as  a  whole ;  the  resulting  judgment  will  be  strong, 
clear,  and  accurate,  in  proportion  to  the  strength,  clear- 
ness, and  accuracy  of  the  united  conceptions.  As  two 
of  the  conceptions  are  increasingly  sharp  and  clear  with 
increasing  distance,  so  does  the  total  resultant  judgment 
increase  in  accuracy  with  increasing  distance. 

It  remains  to  be  said  of  Law  Two,  that  for  very  short, 
or  sub-threshold  distances,  we  should  expect  to  find  its 
effects  very  weak ;  for  them  all  the  elements  of  the  law 
would  be  very  poorly  developed.  The  conception  of  the 
separateness  of  the  points  would  be  faint;  the  skill  of 
measuring  such  unusual  distances  would  be  scanty;  and 
the  immediate  impressions  would  tend  to  group  into 
"spots"  rather  than  to  revive  the  serial  memories  of 
right-lines.  The  judgments  would  be  uncertain,  and 
would,  consequently,  fall  under  the  influence  of  the 
laws  of  uncertainty.  This  brings  us  to  the  law  of 
uncertainty,  which  we  called  Law  Three. 

§  21,  Law  Three,  in  the  number-judgments,  formu- 
lated the  drift  of  the  errors  of  uncertainty.  It  will  do 
the  same  here.  As  before,  the  drift  will  be  toward  the 
average  of  the  possible  categories.  But  the  spatial 
categories  are  more  variously  conditioned  for  different 
regions  of  the  skin,  than  are  the  distance-categories  for 
the  same  regions.  That  is,  the  character  of  the  spatial 
experiences  depend  far  more  on  the  shape,  area,  and 


OUR    NOTIONS    OP    NUIVIBER    AND    SPACE.  97 

contour  of  a  particular  area  of  skin,  or  member  of  the 
body,  than  do  the  numerical  experiences.  For  instance, 
the  distance  series  native  to  the  tip  of  the  tongue 
would  never  be  long,  while  the  number  series  might  run 
as  high  there  as  anywhere.  We  shall  then  expect 
to  find  over-estimates  and  under-estimates  in  any  fixed 
scale  of  categories  of  distance,  like  those  of  our  tables, 
to  be  greatly  variable  between  different  regions  of 
skin.  While  we  shall  find  ourselves  obliged  to  examine 
each  region  more  carefully  by  itself,  to  determine  the 
precise  influence  of  Law  Two  in  each  case,  yet  it  will 
be  just  this  lawful  variableness  which  will  be  of  signifi- 
cance to  us  when,  as  we  propose,  we  come  to  test  the 
truth  of  the  Genetic  Hypothesis,  as  a  Avhole,  by  com- 
parative studies. 

•  §  22.  Beside  the  above  laws,  we  shall  discover  for 
distance-judgments  still  another  law,  which  played  no 
part  in  number-judgments.  We  Avill  call  it  Law  Four, 
and  it  may  be  stated  as  follows,  i.e.,  The  greater  the 
number  of  pins,  the  shorter  will  be  the  estimated  dis- 
tance. And  since  the  right-line  distance  will  be  both 
the  shortest  distance  and  the  correct  distance,  we  may 
say  that,  other  things  being  equal,  by  Law  Four:  The 
greater  the  number  of  pins,  the  more  accurate  should 
be  the  distance-judgment. 

This  is  to  be  accounted  for  as  follows  :  Our  judgments 
are  based  vipon  the  memory  habits  joined  to  particular 


98  OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

nerve-ends.  Also,  our  distance-judgments  are  based 
upon  right-line  movements  between  points.  Now  it  is 
plain  tliat,  in  these  original  movements,  every  nerve 
lying  in  the  line  of  any  movement  would  be  as  much 
joined  to  the  resultant  memor}^  effects  of  that  move- 
ment, other  things  being  equal,  as  would  any  other 
nerve  lying  in  that  line.  Consequently,  although  we 
shall  discover  other  reasons  than  the  above  why  that 
particular  distance-memory,  as  a  whole,  becomes  more 
joined  to  the  end-points  of  the  line  than  to  intermediate 
points,  Ave  still  may  see  from  the  above  reasons  why 
each  intermediate  point  is  very  intimately  joined  with 
that  particular  memory.  This  being  so,  it  is  easy  to 
see  Avhy  every  additional  pin  introduced  between  the 
two  end  pins  in  our  line  of  pins  should  be  an  additional 
stimulant  to  the  revival  of  the  proper  perception  and 
judgment.  Each  pin  in  the  right  line  is  a  guide  toward 
the  distance  perception,  being  based  on  the  right-line 
memory,  ratlier  than  on  the  possible  memory  of  innu- 
merable other  lines.  Consequently,  the  greater  the 
number  of  pins  in  our  experiment,  the  more  accurate 
should  the  judgments  of  distance  be.^ 

iThe  reason  wliy  some  other  category  of  distance  rises  upon 
simultaneous  stimulation  of  two  intermediate  points  of  an  orig- 
inal right-line  movement  (namely,  that  of  the  shorter  distance 
between  the  intermediate  points,  rather  than  the  original  cate- 
goiry)  is  plainly  not  because  the  longer  category  has  no  strength, 
but  because  the  shorter  category  has  the  stronger  and  prevailing 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.  99 

§  23.  Having  explained  our  laws  we  will  now  exam- 
ine our  tables.  Turning  for  illustration  to  Blocks  75  and 
80  of  Table  4,  which  contain  the  figures  corresponding 
to  those  already  used  in  illustrating  the  laws  of  numeri- 
cal judgments,  we  first  observe  in  Block  80  that  all  the 
values  in  the  upper  horizontal  line  are  plus,  and  all  of 
those  in  the  lowest  horizontal  line  (above  the  averages) 
are  minus.  This  is  the  effect  of  "chance,"  and  Law 
One,  which,  it  will  be  observed,  now  works  in  even 
horizontal  lines,  up  and  down,  from  top  to  bottom  of 
the  block,  instead  of  right  and  left,  as  before,  in  the 
number-judgments.  It  would  be  corrected  by  sub- 
tracting a  maximum  amount  from  the  (+)  values  in 
the  top  line,  and  adding  a  like  amount  to  the  (— )  values 
in  the  bottom  line,  and  grading  proportional  corrections 

strength,  as  between  these  two  points.  And  the  reason  why  the 
several  categories,  corresponding  to  each  distance  between  each 
intermediate  point  of  pins,  does  not  rise  to  perception,  under 
simultaneous  impression  of  the  whole  line  of  pins,  is  not  because 
there  is  no  tendency  for  the  several  shorter  categories  to  rise,  but 
because,  again,  the  tendency  for  the  single  longer  category  is  the 
prevailing  category.  Why  the  longer  one  should  be  stronger  than 
the  shorter  ones  is  plain,  if  we  remember  that  each  intermediate 
pin  would  have  some  tendency  to  call  up  the  outside  category, 
while  the  pins  outside  of  each  intermediate  pair  would  not  have 
equal  tendency  to  call  up  the  intermediate  category.  Why  we 
can  think  alone  in  the  one  strongest  category,  and  cannot  think  in 
all  the  categories  at  the  same  time,  lies,  very  probably,  somewhat 
in  the  fact  that  the  same  brain  parts  are  likely  to  be  demanded 
simultaneously  in  the  several  categories,  and  can  act  only  iflu^^JU^^^ 
one  line  of  strongest  tendency.  V-tfe  -i 


<^l 


\ 


<<* 


100       OUR    NOTIONS    OF    NUMBEK   AND    SPACE. 

from  these  extreme  categories  toward  the  middle  cate- 
gory of  3  cm.,  where  the  influence  of  the  law  is 
negative  and  zero. 

§  24.  We  next  observe  the  effects  of  Law  Two.  By 
this,  all  the  judgments  of  distances  above  the  threshold 
distances  should  increase  in  accuracy  with  increase  of 
distance.  Assuming  the  threshold  for  this  region  to 
be  about  3.5  cm.,  we  see  the  number  of  correct  judg- 
ments increasing  regularly  Avitli  increase  of  distance 
throughout  the  remainder  of  the  blocks.  The  average 
in  Block  75,  for  3.5  cm.  is  20.5;  for  5  cm.  is  40.2.  That 
the  laAv  begins  to  have  effect  even  in  the  short  category 
of  1.5  cm.  is  obvious  from  tlie  figvires. 

§  25.  Law  Tliree  shows  a  slight  tendency  toward 
over-estimation  in  the  region  of  the  abdomen  throughout. 
The  total  average  in  Block  80  is  +3.46,  and  the  average 
for  the  middle  category  of  3  cm.  is  -f"  5.6.  The  tend- 
ency is,  however,  so  light,  that  under  the  shortening 
influence  of  Law  Four,  the  actual  tendency  is  toward 
under-estimation  for  V  pins,  even  in  the  short  distances, 
where  by  effect  of  Law  One,  the  actual  judgments  should 
show  plus  values. 

§  26.  The  effects  of  Law  Four  are  most  obvious  in 
the  1-cm.  judgments,  and  in  the  horizontal  averages  of 
Block  75,  and  they  show  markedly  in  Block  80  through- 
out. The  effects  are  exhibited  as  increasing  accuracy 
from  left  to  right,  horizontally,  across  the  four  columns 


OUR    NOTIONS    OF    NU3IBEU    AND    SPACE.       101 

of  pins.  Thus  in  Block  75,  we  read  for  the  1-cm.  judg- 
ments, 42,  53,  66,  78  ;  and  for  the  averages,  30.6,  30.8, 
31.3,  31.6;  and  the  decrease  of  the  amounts  of  error  in 
Block  80  may  be  illustrated  by  the  averages,  which 
read  + 10.3,  +  G-O,  +  •^,  -  2.6. 

§  27.  To  test  the  united  influence  of  our  various 
laws  we  will  now  examine  Blocks  75  and  80  more  inti- 
mately. Studying  the  top  line  of  Block  80,  we  see  as 
folloAvs  :  Law  One  makes  all  tlie  values  more  plus  than 
actually  they  should  be.  Law  Two  shows  a  maximum  of 
uncertainty  for  this,  the  shortest  category  of  distance ; 
the  average  error,  +  26.0,  would  be  the  greatest  for  any 
of  the  distances,  even  after  making  corrections  for  Law 
One.  Law  Three  shows  a  drift  of  error  proportionate 
both  to  the  uncertainties  due  to  Law  Two,  and  to  the 
uncertainties  due  to  Law  Four.  Proportionate  to  Law 
Two,  there  is  over-estimation  holding  good  for  the  aver- 
age of  all  the  minimal  distance-judgments,  the  average 
error  being  -{-  26.0,  —  a  sum  that  indicates  over-esti- 
mation after  correction  for  Law  One.  Proportionate 
to  Law  Four  the  drift  of  error  is  greatest  where  the 
uncertainty  by  Law  Four  is  greatest,  i.e.,  where  the 
pins  are  fewest,  and  decreasing  as  the  pins  increase 
from  left  to  right ;  thus  -f  37.0,  +  29.8,  +  21.0,  +  16.1. 
Law  Four  shows  a  shortening  of  the  judgments  with 
increase  of  pins  ;  this  is  shown  in  the  four  numbers 
last  quoted.     Of  course  the  top  line  of  Block  75  would 


102       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

show  effects  correspouding  to  the  above,  if  correspond- 
ingly analyzed. 

Next,  examining  the  figures  of  column  V  in  both 
blocks,  we  find  :  Law  One  making  the  judgments  too 
long  in  the  short  distances  and  too  short  in  the  long 
distances.  Law  Two  makes  the  judgments  more  accu- 
rate with  increase  of  distance.  This  is  obvious  in  the 
lower  half  of  Block  75,  where  the  number  of  correct 
judgments  increases  with  the  tolerable  regularity  of  20, 
23,  20,  39  for  the  distances  3.5-5  cm.  The  real  effects 
corresponding  to  these,  in  Block  80,  are  obscured  by 
reason  of  the  fact  that  the  minus  quantities  due  to  Law 
One  increase  in  this  column  from  the  middle  down- 
ward, at  a  ratio  greater  than  that  by  which  the  minus 
values,  resulting  from  drift  of  error  (under-estimation  in 
this  V-pin  column)  decrease  through  increase  of  accu- 
racy due  to  Law  Two.  Make  correction  of  Law  One, 
and  Law  Two  is  very  evident.  The  real  effects  of  Law 
Two  in  the  upper  halves  of  the  two  blocks  is  obscured 
in  a  like  manner.  That  is,  in  the  upper  half  of  Block 
80,  we  have  minus  values  even  against  the  influence  of 
Law  One,  which  theoretically  should  give  stronger  plus 
values ;  consequently,  here  there  must  be  under-estima- 
tion by  Law  Three,  heightened  through  the  influence  of 
the  high  number  of  pins,  i.e.,  by  Law  Four.  Now, 
since  the  plus  influence  of  Law  One  falls  off,  with 
increasing    distance,  faster   than   do   the   united   minus 


OUR    NOTIONS    OB^    NUMBER    AND    SPACE.        103 

values  of  Laws  Three  and  Four,  we  therefore  have  the 
apparent  contradiction  of  Law  Two  ;  the  contradiction, 
however,  being  only  apparent,  and  due,  as  we  have  seen, 
to  the  compensating  influences  of  the  other  laws.  The 
influences  of  Laws  Three  and  Four  are,  from  the  fore- 
going, sufficiently  plain,  and  the  entire  distribution  of 
the  figures  in  column  Y  should  now  also  be  clear,  as 
displaying  and  agreeing  with  the  united  influences  of 
our  several  laws. 

Examining  the  lower  horizontal  line  (above  the  aver- 
ages) to  test  our  laws  in  the  higher  distances,  we  find  as 
follows  :  Law  One  gives  minus  errors  throughout.  Law 
Two  gives  maximum  accuracy  for  the  longest  distance  ; 
as  is  evident  from  the  large  number  of  correct  judg- 
ments shown  in  Block  75,  and  the  small  average  error 
that  would  remain  after  correcting  the  minus-constant 
of  Law  One.  Law  Three  averaged  for  the  four  columns 
of  pins,  shows  a  slight  tendency  to  over-estimation  at 
the  distance  of  5  cm..  Block  80.  Law  Four  makes  this 
over-estimation  more  pronounced  in  tlie  low-number 
colvimns,  and  shortens  it  perhaps  to  actual  under-esti- 
mation  in  the  right-hand  or  ••  V-pin  "  column.  As  the 
result  of  the  combined  influence  of  the  four  laws,  the 
effects  of  Law  Four  are,  in  Block  75,  apparently  contra- 
dicted, that  is,  the  actual  number  of  correct  judgments 
decrease  from  left  to  right  (48,  44,  40,  39);  but  it  will 
be    easily    understood    from     the    foregoing    that    this 


104       OUK    NOTIONS    OF    NUMBEll    AND    SPACE. 

appearance  is  but  the  effects  of  the  various  compensa- 
tions which  we  have  last  above  described  and  illustrated 
from  the  last  line  of  Block  75. 

The  distribution  in  the  three  remaining  vertical 
columns  are  so  similar  to  that  of  column  V  that  now, 
having  both  followed  in  detail  tlie  three  sides  of  our 
blocks,  and  also  examined  the  general  influence  of  each 
law  upon  each  block  as  a  whole,  I  think  the  detail  of 
the  distribution  in  all  the  columns  throughout  should  be 
perfectly  plain  to  any  one  upon  due  examination. 

Our  sample  blocks,  therefore,  we  find  conform  to  oiir 
laws.  This  conformation  Avill  have  weiglit  in  support  of 
our  main  hypothesis,  just  in  proportion  as,  with  integ- 
rity, it  may  be  shown  to  be  representative  of  a  wider  con- 
formation extending  throughout  the  large  body  of  our 
experiments.  Manifestly  it  would  be  impossible  within 
any  reasonable  limits  of  publication,  to  go  through  a 
detailed  examination  similar  to  the  above,  explicitly 
demonstrating  the  course  of  our  laws  and  of  our  main 
hypothesis  throughout  each  of  the  123  blocks  of  figures 
presenting  the  extensive  and  arduous  series  of  investi- 
gations classed  together  in  this  paper  as  Experiment 
A.  Still  less  would  it  be  possible  to  extend  such  a 
demonstration  throughout  the  365  blocks  of  Experi- 
ments A  to  E.  This  each  student,  according  to  his 
interest  in  the  matter,  must  do  for  himself.  But  1 
assert  with   the  confidence  of  long  and   careful   study 


OUR    NOTiONS    OF    NUMBER    AND    SPACE.        105 

tliat  such  an  examination  results  in  undeviating  and 
constant  accumulation  of  evidence  of  the  integrity  of 
our  laws  wherever  they  are  implicated,  and  of  the 
soundness  everywhere  of  the  main  lines  of  reasoning 
upon  which  they  have  been  founded.  Where  at  first 
there  may  appear  to  be  contradictions,  we  shall 
discover  by  closer  study,  and  especially  by  the 
comparative  studies  already  foreshadowed,  that  tliese 
are  but  the  ap[)arent  exceptions  which,  when  under- 
stood, all  the  more  abundantly  substantiate  the  general 
truths. 

§  28.  Before  leaving  the  present  discussion  of  dis- 
tance a  few  tilings  remain  to  be  saiil  in  connection  with 
the  special  tables  of  Experiment  A. 

Table  5.  —  Here  the  pins  were  pressed  evenly  without 
rocking.  We  have  already  said,  in  discussing  this  same 
table  under  Number,  that  the  chief  effects  of  the  change 
from  the  method  of  the  regular  experiments  to  the 
present  one,  ought  to  be  a  lessening  of  the  influence  of 
the  laws  most  dependent  upon  present  peripheral  exci- 
tation, and  relatively  to  onliance  the  influence  of  those 
most  dependent  upon  memory. 

Law  Four  is  of  tlie  former  class  ;  its  influence,  there- 
fore, should  be  lessened  in  proportion  as  the  stimulating 
influence  of  each  pin  is  lessened.  Comparing  Block  92 
of  the  new  Table  5,  with  the  corresponding  figures  of 
Block  60  of  the  "regular"  Table  3  for  the  same  region 


106       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

— the  forearm — we  see  that  this  actually  happened.  In 
Block  92  there  is  but  little  shortening  of  the  judgments, 
with  increased  number  of  pins.  The  proper  averages 
now  read  -^  20.4,  +  26.3,  +  25.6,  + 19.2  (indicating 
an  irregular  and  slight  shortening  from  left  to  right 
with  increase  of  pins)  as  against  formerly :  -)- 17.7, 
+  13.3, +9.1,  +7.6  (indicating  a, marked  and  regular 
shortening). 

Law  Three  is  of  the  class  most  exclusively  based  in 
memory  processes  ;  its  effect  by  the  new  method,  there- 
fore, should  be  enhanced,  as  plainly  it  is.  The  average 
error  is  markedly  greater  throughout,  and  the  general 
drift  throws  the  increased  error  more  constantly  toward 
over-estimation.  This  shows  so  plainly  in  the  tables 
that  the  figures  need  not  be  repeated  here.  A  striking 
item  of  confirmation  of  our  interpretation  of  the  com- 
pensating influences  of  the  several  laws,  is  shown  in  the 
lower  line  of  Block  89,  where,  lacking  the  compensating 
influence  of  Law  Four  as  described  in  discussing  the 
corresponding  lines  of  Table  4  on  page  103,  the  number 
of  correct  judgments  no  longer  read  decreasingly  from 
left  to  right,  but  increasingly,  as  lacking  the  influence 
of  Law  Four  they  ought  to  read  (38,  41,  40,  40,  in  Block 
89,  and  57,  40,  49,  48,  in  Block  60). 

Law  Two  being  little  affected  by  the  new  method, 
holds  its  course  even  more  obviously  than  it  did  in  the 
regular  experiments ;    and   this,   because    lacking  in  a 


I 


OUR    NOTIONS    OF    NUMBER    AND    SPACIC.        107 

greater  degree  the  disturbing  influence  of  Law  Four,  is 
what  properly  it  shoukl  do. 

§  29.  Table  7  furnishes  other  peculiar  evidence  for 
our  laws.  It  records  the  results  of  practice.  We  should 
suspect  a  i^riori  that  the  effects  of  practice  would  not 
improve  all  our  laws  of  judgment  equally.  The  law 
which  we  should  most  expect  to  be  improved  by  the 
specific  practice  of  these  particular  experiments  is  Law 
Two ;  this  would  chiefly  result  in  heightening  our  famil- 
iarity with  the  precise  regions  worked  on  ;  which  in 
turn  would  enable  us  to  disassociate  more  sharply  and 
distinctly  the  local  percepts  around  each  pin.  Particu- 
larly would  this  apply  to  our  percepts  of  the  end  pins, 
for  the  reason  that  our  attention  in  judging  distance  is 
proportionally  more  bestowed  upon  these  than  upon  the 
intermediate  pins.  Consequently,  since  improvement 
with  reference  to  the  end  pins  would  mean,  on  the 
whole,  improvement  with  reference  to  the  longer  dis- 
tances, we  should  expect  that  the  chief  consequences  of 
practice  would  be  increased  accuracy  in  judging  the 
longer  distances,  and  more  pronounced  effect  of  Law 
Two  throughout.  Examination  of  Blocks  110  and  llo 
(Table  7)  discovers  this  to  be  just  what  took  place ;  the 
average  error  of  the  longest  distance  is  now  — 5.4  as 
against  —  12.8  formerly  for  the  same  region — the  fore- 
arm—  in  Blocks  55  and  60,  Table  3;  and  the  average 
number  of  correct  judgments  increases  now  for  the  dis- 


108       OUE    NOTIONS    OF    NUMBER    AND    SPACE. 

tances  1  to  3  cm.  by  the  series  of  figures  29.5,  33.0,  39.5, 
37.0,  78.0,  as  against  the  former  series  of  35.7,  44.0, 
47.2,  40.2,  48.5. 

The  specific  practice  with  the  few  distances  actually 
used,  Avould  have  little  direct  influence  upon  the  memory 
habits  of  the  other  distances.  Consequently  the  pull  of 
these  "possible  habits"  upon  the  drift  of  uncertainty, 
where  there  yet  remained  uncertainty,  ought  to  be  about 
the  same  as  before.  The  figures  confirm  this.  Accord- 
ing to  what  we  have  said  just  above  about  the  improve- 
ment in  Law  Two,  we  should  expect  to  find  the  shorter 
distances  remaining  unimproved  as  compared  with  the 
longer  distances.  On  the  average,  their  judgments 
remain  equally  uncertain  with  their  previous  ones. 
The  average  error  for  1  cm.  was  formerly  +  52.4,  and 
after  practice  Avas  +  54.0.  (Separate  experiments  we 
must  not  expect  to  agree  wholly.)  On  the  whole, 
therefore,  these  figures  confirm  what  we  should  have 
expected  as  the  behavior  of  Law  Two,  both  as  to 
average  amount  of  error  and  the  direction  of  its  drift. 

The  matter  is  again  confirmed  when  we  look  at  the 
new  effects  of  Law  Four.  It  was  noted  as  we  became 
more  expert  in  judging  the  distances,  that  we  more  and 
more  based  our  judgments  directly  on  the  impressions 
of  the  end  pins  ;  there  was  less  *' reckoning"  along  from 
pin  to  pin,  such  as  is  based  upon  rocking.  In  other 
words,  the  effects  of  the  intermediate  pins  entered  less 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.        100 

and  less  into  the  judgment,  and  this  is  the  same  as 
saying  that  Law  Four  woukl  have  less  effect  than  before 
practice.  The  obvious  consequence  of  this  ought  to  be, 
in  the  tables,  that  the  judgments  should  exhibit  less 
shortening  than  formerly  from  left  to  right  through 
the  columns  with  the  increasing  number  of  pins,  and 
particularly  this  sliould  be  most  manifest  where  there 
remained  the  greatest  uncertainty.  We  may  now 
observe  that  this  is  precisely  what  did  happen.  Not 
only  do  the  footings  of  Block  113  show  little  of  this 
shortening  as  compared  with  the  footings  of  Block  fiO 
(+ 18.9,  + 19.2,  + 12.2,  +  14.8  now,  and  + 17.7,  +  13.3, 
+  9.1,  +  7.6  formerly^,  but  the  top  lines  of  the  two 
l^locks  show  no  shortening  in  the  1-cm.  judgments  after 
practice,  and  marked  shortening  before  practice  (+69.1, 
+  56.0,  +  44.3,  +  40.0  before,  +  48.0,  +  60.0,  +  55.0, 
+  53.0  after). 

Similar  fulfillment  of  lawful  expectations  can  be  easily 
traced  in  the  number-judgments  of  the  same  special 
experiment.  The  whole  of  Table  7,  therefore,  again 
affords  striking  confirmation  of  our  thesis  in  general. 

§  30.  A  word  must  be  said  of  the  special  experiment 
reported  in  Tables  8  and  9.  If  our  general  discussion, 
and  in  particular  that  part  referring  to  Law  Four,  is 
correct,  then  we  ought  to  expect  that  straight-edges  or 
full  lines  of  cardboard,  pressed  upon  the  skin,  should 
awaken  more  accurate  judgments  than  our  lines  of  pins. 


110       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

It  ought  to  be  a  case  of  Law  Four  with  the  number  of 
pins  raised  to  infinity.  Tables  8  and  9  report  an  exper- 
iment for  testing  this  matter.  In  considering  the 
results  it  should  be  borne  in  mind  that  the  pins  give 
much  sharper  impressions  than  do  the  card  edges. 
Yet,  notwithstanding  that  fact,  the  experiment  is  an 
interesting  confirmation  of  our  general  doctrines.  The 
average  errors  for  the  scale  of  distances  on  the  forearm, 
Block  60,  in  the  regular  experiments  ran  as  follows  : 
+  52.4,  +27.1,  —.1,  —6.9,  —12.8,  while  the  corre- 
sponding errors  with  the  card  edges,  Table  8,  were : 
+  31.4,  +  8.9,  +  2.7,  —  .6,  —  7.8.  Correspondingly  for 
the  number  of  correct  judgments  we  have  for  the  pins  : 
35.7,  44,  47.2,  40.2,  48.5  ;  and  for  the  cards  :  53,  50,  60, 
72,  83.  The  total  averages  are:  ''pins,"  +11.9  and 
43.16,  as  against  "cards,"  +6.9  and  63.8.  Table  9, 
where  the  cards  were  pressed  without  rocking,  also 
yields  its  sliare  of  confirmatory  evidence  of  La,w  Four, 
and  both  Table  8  and  Table  9  are  full  of  points  confirm- 
atory of  our  other  laws,  but  we  have  not  the  space  here 
to  consider  them. 

§  31.  A  glance  at  the  summaries  of  Experiment  A, 
exhibited  in  Table  10,  shows  again  the  integrity  of  our 
laws  in  a  strikingly  impressive  manner,  proportionate 
to  the  extensive  field  of  confirmatory  experimentation 
grouped  into  a  single  view.  Eut  for  the  present  we 
must  leave  our  special  subject  of  distance  to  study 
higher  spatial  complications. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.        Ill 

NUMBER-JUDGMENTS    BASED    ON    TWO 
DIMENSIONS. 

§  32.  In  the  Experiments  B,  C  and  D  our  investiga- 
tions attack  the  psychology  of  two- dimensioned  space. 
Necessarily  Ave  shall  make  but  little  headway  with  it  ; 
our  "  heaps  of  figures  "  will  show  rather  what  in  the 
future  is  to  be  done,  than  reach  complete  demonstration 
of  any  kind. 

Xaturally  we  should  first  inquire  in  this  new  domain, 
whether  the  laws  of  number  and  distance  already  dis- 
covered are  carried  over  into  the  formation  of  the  new 
and  more  complicated  judgments. 

Number. 

§  33.  We  will  first  follow  the  laws  of  number.  For 
these  we  must  study  Tables  11  to  17.  They  relate  to 
Experiment  B,  which  was  conducted  with  pins  arranged 
in  triangles  and  squares,  the  distance  categories  remain- 
ing as  before.^  [The  experiments  subsequent  to  B  do 
not  involve  number-judgments.] 

^  About  these  B  tables  in  general,  a  preliminary  word  of  caution 
is  needed.  By  a  great  fault  not  appreciated  in  laying  out  the 
experiment,  the  highest  category  of  pins  both  with  the  triangles 
and  with  the  squares  was  not  carried  down  through  the  shortest 
distances.  As  a  consequence  there  is  much  complication  in  the 
operations  of  Law  One.  This  is  illustrated  by  any  one  of  the 
blocks,  for  instance  Block  196,  Table  14.     It  will  be  noted  there. 


112       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

That  Laws  Two  and  Three  both  hold  good  throughout 
this  new  set  of  number-judgments,  is  seen  by  slight  ex- 
amination, but  certain  peculiarities,  to  be  observed  here 
and  there  in  the  wider  course  of  general  integrity,  de- 
mand closer  consideration.  Perhaps  the  thing  that  first 
strikes  us  is  the  fact  that  the  number  of  correct  judg- 
ments, and  particularly  in  the  short  distances,  is  on  the 
average  much  greater  than  when  the  pins  were  arranged 
in  a  single  line.  Wliat  has  oi;r  liypothesis  to  say  of 
this  ?  Suppose  three  pins.  A,  B  and  C,  to  be  set  in 
the  corners  of  an  equilateral  triangle  of  1-cm.  base. 
According  to  our  foregoing  discussions,  our  ability  to 
perceive  these  to  be  "  three "  will  depend  upon  the 
predominance,  in  the  combination  of  these  three  par- 
ticular j)oints  of  skin  during  life's  experiences,  of  ''  three 
termed  successions  "  above  all  other  modes  of  eoordi- 
nately  stimulating  them.  What  we  must  now  ask  is 
whether,  other  things  being  equal,  this  mode  of  combi- 
nation is  more  likely  to  occur  when  three  points  are 
arranged  in  a  triangle,  than  when  in  a  straight  line.  Of 
course  it  is  difficult  to  know  Avhen  the  <'  other  things  " 

tliat  the  correct  judgments  in  the  "  Vl-pin  "  cohunn  fall  off  greatly 
in  the  larger  distances  where  the  other  category  of  "  VII  pins" 
has  been  introduced,  from  what  they  were  above  in  the  sh,orter  dis- 
tances. Plainly  this  is  the  effect  of  chance  and  from  opening  a  new 
and  higher  category  which  may  possibly  be  judged.  The  effects 
in  the  averages  are  also  somewhat  disturbed.  With  due  care, 
however,  the  results  may  be  used  comparatively  without  falling 
into  grave  errors. 


OUR    NOTIONS    OP    NUMBER    AND    SPACE.       113 

are  sufficiently  equal  for  just  comparison ;  as,  for 
instance,  with  reference  to  the  distance  between  the 
pins.  In  the  above  supposed  triangle  the  distances 
average  1  cm.  apart.  Perhaps  the  straight-line  category 
coming  nearest  to  this  in  Experiment  A  is  that  where 
the  end  pins  are  1.5  cm.  apart,  and  the  average  distance 
between  the  three  pins  is  1  cm.,  the  same  as  in  the 
triangle.  Assuming  these  two  arrangements  as  concrete 
examples  for  our  comparison,  we  observe  that  the 
triangular  offers  greater  likelihood  for  successive  stim- 
ulation than  does  the  lineal  arrangement.  This  is 
apparent  if  we  consider  the  possible  movements,  in  the 
plane  of  the  skin,  of  a  right  line  which  is  to  be  con- 
sidered with  reference  to  its  stimulations  of  any  three 
given  points  in  the  skin.  If  the  three  points  are  in  a 
straight  line  p,  the  moving  line  I  must  occupy  some 
position  with  reference  to  />  that  can  be  determined  by 
the  angle  x  between  the  lines.  For  our  problem  the 
movements  of  I  must  be  computed  between  the  values 
for  X  oi  0  and  90°.  But  with  a:  =  o,  that  is  when  I  is 
parallel  Avith  p,  all  movements  of  I  over  the  skin  would 
never  be  able  to  stimulate  the  three  points  in  any  sort 
of  succession.  This  state  of  things  could  never  happen 
with  the  points  arranged  in  any  sort  of  triangle.  It  is 
easy,  therefore,  to  show  by  calculation  that,  whether  the 
stimulations  be  made  by  continuous  movements  tan- 
gentially  across  the  skin,  or  by  vertical   pressure   upon 


114       OUll    NOTIONS    OF    NUMBER    AND    SPACE. 

the  skin  in  successively  varying  positions  of  the  stimu- 
lant, the  chances  for  the  proper  successive  combination 
requisite  to  the  development  of  the  threefold  form  of 
numerical  perception  for  the  three  points  would  be 
much  greater  in  life  under  triangular  than  under  lineal 
arrangement.  The  advantage  in  favor  of  the  tri- 
angular arrangement  is  markedly  extended  by  the  very 
important  sort  of  genetic  differentiation  arrived  at 
through  the  principle  of  Concomitant  Variations. 

§  34.  The  empirical  fact  that  our  experiments  show 
the  numerical  judgments  to  be  more  accurate  under 
triangular  than  under  lineal  arrangement  of  the  pins 
would,  therefore,  if  clearly  demonstrated,  be  in  strict 
accord  with  the  theoretical  demands  of  our  general 
thesis  ;  and,  having  the  theory  more  fully  before  us,  we 
must  now  examine  this  demonstration  in  our  tables 
more  particularly  theretoward. 

Assuming  that  the  "  Ill-pin,  l.o-cm."  results  pf  Ex- 
periment A  may  be  compared  with  the  "  Ill-pin,  1-cm." 
results  of  Experiment  B  —  an  assumption  which,  as  we 
will  presently  show,  favors  the  right-line  arrangement  — 
we  get  from  the  several  tables  the  following  averages 
for  number  of  correct  judgments  and  for  average  error. 
The  figures  in  the  left-hand  column  refer  to  the  lineal, 
those  in  the  right-hand  to  the  triangular  arrangement  : 


OUK,    NOTIONS    OF    NUMBER    AND    SPACE.       115 


Toncfiie 


Forehead 


Forearm 


Abdomen 


Block    5 
"       10 


Block  25 
"       30 


Block  45 
"       50 


Block  Go 
"       70 


Line. 


99 

+ 

.o 

27 

+ 

24.8 

Tkiajnole. 


34 
+     9.3 


19 
+  39.8 


98 
+   .6 

Block  128 
"   133 

20 
+  49.3 

Block  148 
153 

28 
+  38.7 

Block  173 
"   178 

4(; 

+  32.0 


Block  190 
199 


The  above  figures  are  given  without  correction  being 
made  for  Law  One.  To  make  such  corrections  w^e 
should  have  to  subtract  much  larger  amounts  from  the 
"  triangle  "  results  than  from  the  "  line  "  results  ;  we 
should  liave  to  do  this  because  of  the  difference  in 
position  relative  to  Law^  One  of  the  "  Ill-pin  "  column 
in  the  two  cases.  I  will  not  attempt  the  proper  correc- 
tions, but  it  will  be  evident  to  any  one  upon  due 
consideration  that,  within  very  safe  estimates,  they 
would  demonstrate  conclusively  the  greater  accuracy  of 
the  judgments  under  the  triangular  arrangement  than 
under  the  lineal. 

Upon  the  same  basis  of  average  distances,  if  we  com- 
pare the  3-cm.  judgments  of  Experiment  A  with  those 
at  2  em.  in  Experiment  B,  we  get  the  following  : 


116       OUE    NOTIONS    OF    NUMBER    AND    SPACE. 


Line. 

Tkiancli;. 

Tongue 

S   Block  5 
/    "   10 

99 
+   .3 

99 
+   .2 

Block  128 
"   133 

rorehead 

\      Block  25 
1    "   30 

00 
+  4.3 

50 
+  29.8 

Block  148 
"   153 

Forearm 

\      Block  45 
(    "   50 

27 
+   .9 

40 

+  28.0 

Block  173 

"   178 

Abdomen 

(   Block  05 

1    "70 

28 
+  29.1 

50 
+  24.0 

Block  190 
"   199 

These  figures  also,  when  proper  corrections  sliould  be 
made  for  Law  One,  would  show  the  superiority  of  the 
triangle  judgments  to  be  very  marked. 

§  35.  It  should  now  be  noted  that  the  above  method 
of  averaging  the  distances  is  very  unfair  in  favor  of  the 
lineal  arrangement.  The  judgments  at  2  cm.  ar,e  better 
than  at  1  cm.,  and  manifestly  it  would  not  be  right  to 
average  100  of  the  former  against  200  of  the  latter. 
But  this  is  practically  what  we  do  in  the  above  method 
when  we  make  the  2  cm.  of  distance  between  the  two 
end  pins  in  the  lineal  arrangement  offset  two  distances 
of  1  cm.  each  in  the  triangular  arrangement.  This 
should  be  borne  in  mind  in  making  corrections  in  the 
two  above  tables  and  for  the  comparisons  now  in  hand 
all  through. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       117 

§  36.  80  much  for  tliree  pins.  A  similar  compari- 
son with  the  foregoing  may  now  be  made  for  the  "fonr- 
pin"  judgments.  For  this,  by  the  above  method  of 
simply  averaging  the  inter-distances,  we  should  have  to 
compare  the  1-cm.  judgments  of  four  pins  in  a  square, 
with  the  2.5-cm.  judgments  of  four  pins  in  a  line,  and 
such  violent  errors  would  arise  from  offsetting  2.5-cm. 
judgments  with  two  and  a  half  times  as  many  1-cm. 
judgments  as  to  make  such  comparisons  wholly  inad- 
missible. AVe  can,  however,  arrive  at  the  desired 
information  by  a  more  satisfactory  method.  In  Table 
17,  Block  210,  we  find  the  total  average  of  correct  judg- 
ments summarized  for  all  the  regions  of  skin  worked 
on,  to  be:  for  ''III  pins,"  48.95,  and  for  ''IV  pins," 
54.60.  In  Block  217,  the  corresponding  average  error 
for  ''III  pins"  is  +23.2;  for  "IV  pins,"  +17.0.  As 
the  conditions  of  Law  One  were  precisely  similar  for 
these  two  sets  of  figures,  they  show  conclusively  that 
four  ])ins  in  a  square  are  more  accurately  judged  tlian 
three  q)ins  in  a  triangle.  We  have  already  demonstrated 
in  our  experiments  that  three  pins  in  a  triangle  are 
judged  better  than  three  in  a  line.  To  complete  our 
proof,  tlierefore,  that  four  in  a  square  are  judged  more 
accurately  than  four  in  a  line,  we  have  but  to  show  that 
three  in  a  line  are  better  judged  than  four  in  a  line.  It 
is  true  that  the  general  summary  of  the  line  experi- 
ments, Table  10,   Block  118,  shows  49.2   correct  judg- 


118       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

ments  for  "IV  pins,"  and  only  37.7  for  "III  pins."  But 
before  we  take  this  as  evidence  tliat  four  pins  are  better 
judged  than  three,  we  must  again  make  the  proper 
corrections.  We  must  remember  that  all  through 
Experiment  A  we  found  over-estimation ;  that  the  posi- 
tion of  the  "IV-pin"  column  under  Law  One  apparently 
offset  this  general  over-estimation,  making  the  "IV-pin" 
judgments  appear  unduly  accurate,  Avhile  the  position  of 
the  "Ill-pin"  column  augmented  the  over-estimation, 
and  heightened  the  errors.  The  figures  as  given,  there- 
fore, show  an  illusive  comparison.  The  illusion  is 
greatest  where  there  is  the  greatest  over-estimation, 
namely,  in  the  shorter  distances.  It  will  be  observed 
that  the  longer  distances  throughout  give  undoubted 
evidence,  even  without  proper  corrections,  that  three- 
pins  are  fundamentally  judged  with  greater  accuracy 
than  four,  the  arrangements  being  the  same.  But  mak- 
ing the  proper  allowances  for  the  compensations  between 
Law  One  and  Law  Two,  and  the  fact  which  common 
sense  would  assert  from  the  outset,  that  in  similar  lineal 
arrangement  three  pins  are  judged  more  easily  than 
four,  will,  I  think,  receive  from  the  figures  of  Experi- 
ment A  throughout  most  unmistakable  demonstration. 
This  being  so,  it  is  therewith  also  demonstrated 
that  four  pins  in  a  square  are  judged  better  than  four 
pins  in  a  line,  which  was  the  main  proposition  under 
consideration. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.        119 

§  37.  In  the  above  paragraph  I  have  incidentally  stated 
the  important  fact  brought  out  by  Experiment  B,  that 
four  pins  in  a  square  are  judged  better  than  three  in  an 
equilateral  triangle  of  the  same  base.  Why  this  should 
be  so  under  our  hypothesis,  Avhile  "common  sense" 
would  expect  the  contrary,  I  have  space  here  to  demon- 
strate only  partially.  We  may  anticipate  that  we  have 
to  do  here  with  matters  of  experience  conditioned  by 
geometric  arrangements.  By  Law  Two,  pairs  of  points 
are  increasingly  disassociated  proportionally  to  their 
distance  apart.  The  diagonal  points  of  the  square  are 
further  apart  than  any  pair  of  points  in  the  triangle. 
By  the  mere  law  of  distance-average,  therefore,  the 
square  should  rank  above  the  triangle.  Ko  doubt  the 
diiference  between  the  lengths  of  the  hypotenuse  and 
the  sides  of  the  square  give  a  favorable  "  cue  "'  in 
reasoning  out  that  the  figure  pressed  on  the  skin  must 
be  a  square  and  therefore  has  four  pins.  But  we  may 
note  as  to  this,  that  while  the  difference  of  length 
between  the  inter-distances  in  that  category  of  our 
experiment  where  four  pins  are  arranged  in  a  triangle 
— one  being  in  the  center — is  greater  than  in  the  four- 
I)in  square,  yet  the  judgments  for  such  an  arrangement 
of  four  pins  are  inferior  to  those  of  the  square  of  the 
same  base.  (Block  216,  IV  pins  in  triangle  47.20,  in 
square  54.60.)  We  must  consider,  therefore,  that  the 
superiority  of  the  square  over  the  triangle  in  numerieaJ^:^," 


ih  A  *>; 


120        OUK    NOTIONS    OF    NUMBER    AND    SPACE. 

judgment,  is  fundamentally  rooted  in  tlie  integrity  of 
Law  Two  carried  over  into  the  more  complicated  judg- 
ments of  two-dimensioned  space.  In  this  light  the 
whole  matter  becomes  confirmatory  of  our  genetic 
hypothesis,  and  highly  instructive  as  to  the  intimate 
formation  of  those  mental  processes  which  are  a  degree 
more  complex  than  the  most  simple  ones. 

§  38.  As  bearing  on  the  above  I  have  only  room  to 
note  in  Table  15,  Avhere  no  rocking  was  permitted,  that 
the  IV-pin  square  still  shows  superior  to  the  Ill- 
pin  triangle,  while  in  Table  16,  where  the  mind  was 
forced  to  neglect  the  geometric  impressions  and  to  attend 
alone  to  the  points  actually  felt,  the  averages  show  the 
III  pins  to  be  judged  correctly  9.2  times  in  the  tri- 
angle, and  the  IV  pins  only  8.6  times  in  the  square. 
That  is,  when  the  geometric  influence  is  shut  out,  the 
judgments  fall  back  to  the  common  principle  that  three 
impressions  may  be  better  distinguished  than  four, 

§  39.  Turning  from  Law  Two  to  Law  Three  we 
discover  upon  slight  study  that,  within  categories  as 
nearly  similar  as  could  be  chosen  for  comparison  of  the 
different  arranging  of  the  pins,  the  same  general  drift 
of  over-estimation  is  manifested  throughout  Experiment 
B  as  we  discovered  throughout  Experiment  A  ;  also, 
over-estimation  is  greatest  now  in  the  same  relative 
places  as  formerly,  namely,  in  the  shorter  distances. 
These    above    facts    are,    perhaps,    evidence    for    laws 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       121 

already  redundantly  confirmed,  but  the  subject  gains 
extended  interest  when  we  examine,  from  another  point 
of  view,  the  amount  and  the  distribution  of  the  errors 
in  the  B  experiments. 

We  could  have  expected  from  Laws  One  and  Three, 
that  the  greater  the  number  of  numerical  categories 
used  in  any  experiment,  the  greater  would  be  both  the 
amount  and  the  drift  of  the  errors  made  under  these 
two  laws.  For  instance,  if  we  made  new  investigations 
like  those  of  Experiment  A,  but  with  numbers  of  pins 
ranging  from  III  to  IX  (as  in  Experiment  B),  in  place 
of  from  II  to  Y,  as  formerly,  we  should  expect  the 
amount  and  the  range  of  errors  to  be  much  increased. 
Upon  the  face  of  it,  the  possibility  of  error  where  the 
judgments  may  range  from  III  to  IX  is  greater  than 
when  they  are  limited  between  II  and  V  ;  both  the 
mathematical  chances  are  greater  under  Law  One,  and 
the  psychological  chances  are  greater  under  Law  Three. 
When,  however,  we  study  the  results  of  Experiment  B, 
we  find  quite  the  reverse  of  what,  from  the  above,  was 
to  have  been  expected. 

By  way  of  examining  into  this  Ave  must  first  grasp 
more  clearly  the  relative  amounts  of  error  made  in  tlie 
two  experiments.  We  had  trouble  in  getting  an  exact 
basis  for  this  comparison,  but  in  rough  ways  we  may 
yet  get  truer  ideas  of  it.  All  the  conditions,  save 
those  whose  results  we  are  seeking  to  measure,  favor 


122       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

the  Il-pin  judgments  ;  that  is,  other  things  equal,  II 
pins  ought  to  be  judged  better  than  III.  If,  now, 
the  new  triangle  judgments  happen  to  exhibit  a  less 
amount  of  error  than  did  the  old  Il-pin  judgments,  we 
may  then  justly  take  the  amount  of  this  improvement 
to  be  a  partial  measure  of  the  new  conditions  of  the 
experiment.  For  a  full  comparison  one  must  go  to  the 
full  tables,  but  we  will  bring  forward  a  few  test  items. 
In  the  following  table  we  will  compare  both  the 
maximum  amounts  of  error  made  for  II  pins  in  Experi- 
ment A  with  those  for  II  pins  in  Experiment  B,  and 
also  the  corresponding  average  errors: 


Similarly,  the  total  average  in  the  general  summaries 
(Tables  10  and  17)  are  for  II  pins  +  41.6,  and  for  the 
Ill-pin  triangle  +23.2.  For  a  still  rougher  comparison, 
the  grand  averages   of  the  same  summaries  give  us  the 

1  See  upper  left-hand  corner  of  Block  70,  Table  4,  and  cor- 
respondingly for  other  tables. 

2  See  footing  of  left-hand  column,  Block  70,  Table  4,  and  cor- 
respondingly for  other  tables. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       123 

following  errors:  Experiment  A,  +  5.8  ;  Experiment 
B,  +2.1  (lower  left-hand  corner,  Blocks  121  and  217). 
The  superiority  of  the  triangle  arrangement  is  obvious 
everywhere  without  comment. 

§  40.  The  above  figures  having  furnished  us  a 
clearer  demonstration  of  the  amount  of  superiority  of 
two-dimensioned  judgments  over  the  lineal,  we  must 
look  at  certain  differences  in  the  distribution  of  these 
errors  under  the  two  experiments.  We  get  at  these 
quickest  by  an  illustration.  If  we  turn  back  to  our  old 
typical  Block  65  and  read  the  top  line,  we  get  the  cor- 
rect judgments  at  1-cm.  distance  for  II,  III,  IV  and  V 
pins,  respectively,  as  follows:  7,  32,  59,  55.  We 
remember  the  explanation  of  this  remarkable  increase 
of  apparent  accuracy  from  left  to  right;  that  it 
Avas  due  to  the  drift  of  the  great  uncertainty  of  the 
short  distance-judgments.  If  now  we  turn  to  the  top 
line  of  the  corresponding  Block  196  of  Table  14,  we 
get,  with  a  similar  ascending  series  of  pins,  a  descending 
series  of  correct  judgments,  as  follows:  For  triangles 
46,  32,  31;  and  for  squares  44,  19,  28.  We  have  no 
longer  tlie  remarkable  increase  from  left  to  right  due  to 
drift  of  error  under  Laws  One  and  Three.  What  is  the 
trouble  ?  Are  these  laws  inoperative  here  ?  Xo  !  but 
we  have,  in  a  particular  but  legitimate  exception,  a 
remarkable  proof  of  the  general  integrity  of  these  laws 
everywhere,  both  throughout  Experiment  A  and  tlirough- 


124       OUR    NOTIOISS    OF    NUMBER    AND    SPACE. 

out  Experiment  B.  We  have  recalled  that  the  drift  from 
left  to  right  in  Block  65  was  a  drift  of  uncertainty).  If 
now  we  look  at  the  top  line  of  Block  199  we  see  thajb 
the  uncertainty  there  is  very  small  as  compared  with 
the  corresponding  line  of  Block  70.  The  maximum 
error  is  for  the  former  +32.0;  for  tlie  latter  +109.9. 
The  average  error  for  the  former  is  —  .3  ;  for  the  latter 
+  35.7.  The  drift  of  error  from  left  to  right  in  the 
new  experiment  ought,  therefore,  to  be  proportional  to 
this  marked  decrease  in  error.  And  so  it  is.  Appar- 
ently there  is  none  whatever,  and  the  number  of  correct 
judgments  actually  decrease  from  left  to  right.  It  is 
possible  that  corrections  for  Law  One  would  still  leave 
a  small  drift  under  Law  Three,  but  in  any  case  it  would 
be  so  small  as  to  constitute  a  peculiar  proof  of  the 
integrity  of  this  law  carried  up  into  the  more  compli- 
cated judgments  of  two-dimensioned  space.  Of  course 
this  is  but  a  sample  of  proof  which  would  be  augmented 
shovild  we  extend  our  examinations.  For  the  forehead 
and  for  the  forearm  the  amount  of  error  is  less  in 
Experiment  B  than  in  A,  but  not  so  much  less  propor- 
tionally, as  in  the  above  sample  taken  from  the  abdomen ; 
accordingly,  the  drift  under  Law  Three  should  be  less 
than  formerly,  but  not  so  much  so  as  in  our  above 
illustration.  This  is  just  what  occurred,  as  the  follow- 
ing figures  taken  from  the  tables  will  demonstrate. 
They  give  the  top  lines  respectively  of  Blocks  25,  30, 
45,  50,  148,  153,  173  and  178  : 


OUR   NOTIONS    OF    NUMBER    AND    SPACE.       125 


Experiment  A. 

Forehead. 

5 

+  84.2 

16                  61 
+  43.5         +     1.5 

Forearm. 

64 
-    8.7 

26 
+  70.0 

31                 45 
+  18.1         -  11.7 

Experiment  B. 

Forehead. 

23 
-  27.3 

26 
+  69.1 

28 
+  35.0 

54                    27 
+  12.9           +  53.5 

Forearm. 

54 

+  28.9 

78 
-    8.9 

28 
+  58.2 

28 
+  21.4 

34                    34 
-    9.5           +  32.4 

35 
+  10.3 

44 
-  21.5 

Here,  for  the  forehead  in  Experiment  A,  between  II 
and  V  pins,  we  see  a  drift  of  correct  judgments  from 
5  to  64,  and  of  amount  of  error  from  +84.2  to  — 8.7  ; 
while  in  Experiment  B  the  corresponding  drift  between 
III  and  IX  pins  is  only  26  to  78,  and  +69.1  to  —8.9; 
and  so  similarly  for  the  forearm. 

§  41.  Had  we  space,  we  ought  to  consider  further  the 
distributing  influences  of  our  laws  under  the  different 
conditions  of  our  experiments  and  for  the  different 
regions  of  the  body;  but  this  we  must  now  leave  to  the 
individual  student.  Compelled  now  to  pass  on  to  other 
matters,  we  may  summarize  our  imperfect  study  of  the 
number-judgments  in  our  new  Experiments  B,  as  fol- 


126       OUE    NOTIONS    OF    NUMBER    AND    SPACE. 

lows:  We  observe  in  the  new  workings  of  our  laws 
botli  certain  modifications  of  old  traits  running  parallel 
to  definite  changes  in  the  conditions  under  which  they 
act,  and  also  entirely  new  traits  due  to  new  conditions. 
We  find  all  these  manifestations  agreeing  with  each 
other  and  conforming  to  the  reasonings  of  our  general 
hypothesis;  we  must,  therefore,  admit  them  to  be  strong 
evidence  for  its  truth. 


DISTANCE-JUDGMENTS    BASED    ON    TWO 
DIMENSIONS. 

(^Experiments  B,  C  and  D.) 

§  42.  The  elementary  law  of  Association  is  that 
the  resultant  state  at  any  moment  is  the  indissoluble 
product  of  the  sum  of  all  the  tendencies  active  at  that 
moment.  It  is  fundamental  to  all  that  I  have  hcBetofore 
said,  that  this  law  holds  good  for  tlie  stimulation  of 
each  and  every  possible  combination  of  nerve-ends. 
When  we  stimulate  a  single  nerve  we  get  a  specific 
effect  which  expresses  the  tendencies  developed  for 
that  definite  stimulation.  AVhen  two  given  nerves  are 
stimulated  we  get  a  different  effect,  also  specific,  and 
expressing  the  tendencies  developed  coordinately  for 
the  given  combination  as  stimulated.  Just  what  rela- 
tionship the  specific  state  (which  expresses  the  sum  of 


OUR    NOTIONS    OP   NUMBER   AND    SPACE.       127 

combined  stimulation)  bears  to  the  several  specific 
states  (which,  respectively,  on  occasions  express  the 
stimulation  of  each  nerve  separately),  our  science  does 
not  now  with  confidence  suggest.  We  cannot  yet  for- 
mulate the  coordinate  tendency,  in  terms  of  tlie  several 
separate  tendencies.  But  while  we  may  not  determine 
its  particulars,  we  may  demonstrate  that  there  is  such 
a  relationship ;  and  this  is  to  be  my  present  thesis. 
Stated  more  explicitly  it  is  that :  The  actual  effect  of 
any  combination  of  nerves  is  partially  the  resultant  of 
the  combined  experiences  of  the  given  combination,  and 
partially,  also,  is  to  be  traced  back  to  the  experiences 
which  have  influenced  and  developed  each  element  or 
possible  sub-combination  of  elements  in  the  total 
combination  separately.  It  is  the  latter  part  of  this 
statement  which,  in  approaching  more  complicated 
distance-judgments,  we  must  especially  consider.^ 

§  43.  We  may  profitably  bring  this  matter  before  us 
by  considering  one  of  the  triangles  of  Experiment  C. 
The  sides  of  these  triangles  were  formed  by  straight 
edges  of  cardboard,  the  card  being  folded  as  when 
forming  the  sides  of  a  paper  box.  We  are  to  study  the 
coordinated  result  of  pressing  one  of  these  lineal  figures 
iipon  tlie  skin.     C-all  the  triangle  ABC.     ISTow,  by  our 

1  It  must  be  borne  in  niiud  that  we  never  suggest  that  the 
resultant  mental  state  is  other  than  an  indecomposable  specific 
wliole. 


128       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

thesis,  upon  the  simultaneous  stimuLation  of  the  lines 
A,  B,  C,  there  will  tend  to  rise  every  effect  common  to 
the  separate  stimulation  of  every  s\ib-combination  of 
nerve-ends  possible  under  the  laws  of  permutation  and 
combination  for  the  total  number  of  nerve-ends  in  the 
wliole  lineal  triangle.  We  do  not  say  that  the  actual 
effect  will  be  the  resultant  solely  of  the  sum  of  these 
tendencies.  On  the  contrary,  every  time  the  whole 
triangle  is  affected,  as  in  our  present  case,  there  will 
be  left  thereby  a  direct  modification  of  the  habit  of 
reaction  of  the  total  combination,  and  this  modification 
will,  in  some  degree,  manifest  itself  in  all  subsequent 
activities  of  the  total  combination.  But  we  are 
now  to  consider  the  tendencies  of  the  many  possible 
sub-combinations. 

If  we  examine  the  case  of  any  two  points,  y  and  x, 
in  the  perimeter  of  the  triangle  (one,  at  least,  of  the 
same  not  being  a  corner)  we  see,  by  our  general,  thesis, 
that  the  distance-judgments  based  upon  the  separate 
stimulations  of  any  such  points  would  be  shorter  than 
that  of  the  sides  of  the  triangle.  Consequently,  by  our 
present  thesis,  the  influence  of  all  such  tendencies,  in 
the  sum  of  tendencies  resultant  from  stimulating  the 
whole  triangle  coordinately,  ought  to  shorten  any  dis- 
tance-judgment simultaneously  based  thereupon.  If 
this  resultant  judgment  is  to  estimate  the  length  of  the 
sides  of  the  triangle,  or  of  one  particular  side,  then, 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       129 

although  the  "developed  tendency"'  corresponding  to 
the  side  may  be  the  chief  factor  in  the  formation  of  the 
judgment,  yet  these  other  tendencies,  being  in  activity 
by  reason  of  actual  peripheral  stimulation,  they  cannot 
be  wholly  got  rid  of  and  will  enter  into  the  sum  total  of 
present  influence ;  and,  being  shorter  than  the  tendency 
corresponding  to  the  full  side,  they  will  shorten  the 
resultant  judgment  from  what  it  would  be,  were  the 
same  distance  measured  simply  between  two  pins.  If, 
thus,  our  thesis  is  correct,  all  the  distance-judgments 
for  triangles  in  Experiment  C  ought  to  be  shorter  than 
judgments  of  corresponding  distances  in  Experiment  A. 
§  44.  Before  we  examine  our  tables  to  test  this 
theoretical  conclusion,  a  relative  matter  must  be  con- 
sidered. It  concerns  the  ''sharpness"  of  the  different 
modes  of  stimulation.  Up  to  the  present,  to  avoid  con- 
fusion, I  have  excluded  the  factor  of  ''pure  sensibility" 
from  our  studies.  In  proper  place,  I  shall  bring  in 
special  investigation  to  demonstrate  what  I  shall  state 
here  dogmatically,  namely,  that,  within  certain  limits, 
^'sharpness  of  stimulation^^  shortens  the  resultant  dis- 
tance-judgment. Xow,  according  to  this,  and  since  the 
pin-points  of  Experiment  A  furnish  a  much  sharper 
mode  of  stimulation  than  the  card-edges  of  Experiment 
C,  in  comparing  the  judgments  of  the  two  we  must  bear 
in  mind  that  those  in  Experiment  A  are  shorter  than, 
for  a  perfectly  just  comparison,  they  should  be.     If  in 


130       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

tlie  actual  results  the  theoretical  demands  are  some- 
times apparently  unfulfilled,  due  allowance  must  be 
made  ;  and  if  they  are  fulfilled  without  such  allowance, 
the  proof  of  our  thesis  must  be  considered  as  the  more 
marked. 

§  45.  Toward  making  the  proper  .comparisons,  I  first 
present  the  following  table.  In  the  vertical  columns  are 
given  alternately,  the  maximum  and  the  average  over- 
estimation  taken  correspondingly  from  the  '■'■average'''' 
distance-judgments  of  Experiment  A,  and  the  triangle 
distance-judgments  of  Experiment  C.  I  have  used  the 
"average"  here  in  place  of  the  Il-pin  judgments  referred 
to  in  the  theoretical  discussion,  as  it  may  be  claimed  that 
the  latter  are  unduly  lengthened  by  the  distributing 
influence  of  Law  One.  The  "averages,"  since  they  are 
free  from  such  criticism,  are  in  theory  equally  eligible 
to  the  proper  comparison  and,  in  fact,  are  shorter,  as,  by 
Law  Four,  they  should  be,  and  will,  therefore,  if  they 
stand  the  test,  demonstrate  our  point  in  hand  even  more 
profitably  than  would  the  "Il-pin"  judgments: 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       131 


H 


+  I 


+  + 


O  CO 
O  lO 

++ 


+  + 


o  c; 
cc  -^ 

+  + 


c  o 

^  CO 
+  + 


+  + 


+  + 


lO 

o 

. :         - 

2^ 

CO 

o 

oc 

^ 

,« 

^ 

r« 

o 

o 

o 

o 

o 

o 

o 

o 

pq 

W 

M 

K 

^ 

^ 

^ 

^ 

<N 

CO 

'^ 

o 

(U 

(U 

a> 

« 

^ 

^ 

J2 

.a 

c« 

eS 

eS 

ce 

H 

H 

H 

H 

^^ 


132       OUR    NOTIONS    OF    NUMBER   AND    SPACE. 

These  figures,  with  the  exception  of  those  for  the 
abdomen,  confirm  our  theory,  even  without  allowance 
being  made  for  sharpness.  And  as  the  abdomen, 
owing  to  the  thinness  and  tenderness  of  the  skin, 
is  just  the  region  where  sharpness  of  the  impressions 
from  the  pins  would  be  of  greatest  effect,  as  compared 
with  the  rather  dull  feeling  from  the  paper  triangles, 
I  think  the  evidence  of  the  table  is  beyond  question. 
Particularly  we  call  attention  to  the  figures  for  the 
last  line  of  the  table  ;  these  are  for  the  ''  (a)  method," 
where  the  apparatus  is  applied  evenly,  without  rocking. 
There  is  reason  to  believe  that  the  effects  under  present 
discussion  would  come  out  purest  in  this  method.  It 
is,  therefore,  of  interest  to  note  that  the  contrast  of 
the  two  sorts  of  judgments  is  more  marked  in  these 
results  than  anywhere  else  in  the  table. 

§  46.  The  straight-edge  judgments  of  Experiment  A 
(see  Tables  8  and  9,  discussed  on  page  109)  were^  shorter 
than  those  for  the  same  distances  measured  by  pins  in 
straight  line.  Theoretically,  for  the  reasons  already 
given,  our  triangle-judgments  should  be  shorter  than 
either.  Comparing  the  average  errors,  respectively, 
from  Tables  8  and  9,  with  the  corresponding  figures 
for  triangles  in  Tables  19  and  21,  we  get  as  follows  : 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       133 


Experiment  A. 

Experiment  C. 

Straight-Edge.        j 

Triaxglk. 

i 

Regular  Method  .  . 

+    G.9 

+    5.3 

Method  (a) 

+    8.9 

+    2.4 

Again,  without  allowance  for  sharpness,  the  average 
errors  all  show  the  superiority  of  the  triangle-judg- 
ments, and  again  the  contrast  is  most  pronounced  in 
the  purer  (<i)  method.  But  allowance  for  sharpness 
should  be  made  even  here.  True,  the  straight-edges 
were  made  of  paper,  as  well  as  the  triangles.  But, 
with  the  former,  where  the  hard  cardboard  is  cut 
squarely  off,  the  corners  are  nearly  as  sharp  as  pins, 
whereas  the  same  cards,  when  fitted  together  as  per- 
fectly as  possible  into  the  obtuser  figures  of  Experi- 
ment C,  make  corners  that  are  far  duller  and  more 
indistinct.  On  the  whole,  therefore,  these  figures  speak 
pronouncedly  the  superiority  of  the  triangle  judgments 
over  even  those  of  single  lines. 

§  47.  But  we  now  come  to  evidence,  in  bulk  and 
unmistakable  clearness  confirmatory  of  our  theories, 
outweighing  all  that  Ave  have  previously  reached.     For 


134       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

reasons  so  similar  to  those  above  given,  for  the  superi- 
ority of  the  triangle-judgments  over  those  for  single 
lines,  that  they  may  be  here  omitted  in  detail,  it  is  de- 
manded by  our  hypothesis,  other  things  being  equal,  tliat 
the  sides  of  squares  should  be  judged  to  be  longer  than 
the  sides  of  triangles  of  equal  length.  Moreover,  the 
diameter  of  circles  should  be  judged  to  be  less  than  the 
sides  of  squares,  and  to  be  greater  than  the  sides  of 
triangles,  the  real  distances  all  being  equal.  Experiment 
C  was  specially  designed  to  investigate  these  matters, 
and  the  results  are  among  the  most  conclusive  which 
I  have  to  present.  Throughout  the  whole  set  of  C 
experiments,  both  the  figures  for  number  of  Correct 
Judgments  and  those  for  Amount  of  Error,  balancing 
themselves  within  small  margins  of  variation,  demon- 
strate, in  accordance  with  our  thesis,  the  proper 
relationships  of  accuracy  and  foreshortening,  respec- 
tively, as  between  triangles,  circles  and  squares,  jn  a  way 
that  is  truly  remarkable  ;  is  the  more  remarkable  since 
the  conditions  are  here  perfect  for  comparison  through- 
out— no  allowances  or  corrections  having  to  be  made 
for  anything.  We  need  not  repeat  here  any  figures  for 
the  results,  for  they  are  everywhere  plainly  to  be  under- 
stood, upon  reference  to  any  of  the  Tables  18a  to  22a, 
inclusive.  The  triangles  are  always  judged  shorter 
than  the  circles,  and  the  circles  shorter  again  than  the 
squares.     The  importance  of  such  a  mass  of  testimony. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.        135 

however,  must  not  be  underrated  because  of  the  brevity 
in  which  it  can  be  mentioned.^ 

§  48.  But  important  as  this  mass  of  evidence  is, 
we  find  it  duplicated  in  the  several  tables  of  Ex- 
periment D  (18b  to  22b,  inclusive).  Throughout  these, 
we  again  find  the  sides  of  triangles  judged  to  be 
shorter  than  the  diameters  of  circles,  and  the  latter 
to  be  shorter  than  the  squares  ;  for  all  of  which,  as 
before,  we  refer  directly  to  the  tables. - 

§  49.  Experiment  B  furnishes  another  mass  of 
evidence.  Here  we  have  no  circles,  but  the  arrangement 
of  four  different  number  categories,  similarly  in  triangles 
and  in  squares,  gives  opportunity  for  instructive  varia- 
tions.    I  will  first  ask  you  to  compare  the  averages  of 

1  In  studying  the  tables  for  C  and  D  the  figures  for  the  number 
of  correct  judgments  must  be  interpreted  with  care.  For  instance, 
these  respectively  for  triangles,  squares  and  circles,  read  in  Block 
226,  for  the  top  line  86,  32,  72  ;  and  for  the  bottom  line  48,  88,  62  ; 
yet  both  are  precisely  what  they  should  be  to  demonstrate  our  thesis 
throughout.  It  mu.st  be  recalled  that  Law  One  operates  here  as 
elsewhere.  If  we  glance  at  Block  229,  showing  the  corresponding 
amounts  of  error  for  the  above  judgments,  we  see  the  explanation 
of  them  at  once:  In  the  top  line  there  is  undue  over-estimation 
throughout  (Law  One) ;  consequently  the  relatively  shorter  judg- 
ments of  the  triangles  are  more  frequently  right  than  the  squares 
or  circles ;  hence  the  figures  86,  .32,  72.  In  the  bottom  line  there 
is  undue  foreshortening  throughout  (Law  One ) ;  consequently  the 
triangles  here  are  less  frequently  right  ;  and  hence  the  figures  48, 
88,  62.  The  relative  length  of  triangles,  squares  and  circles  is 
perfect  everywhere. 

■-  It  should  be  recalled  here  that  for  the  D  apparatus  the  figures 
are  made  of  cork  and  present  a  smooth,  even,  solid  surface  to  be 
pressed  upon  the  skin. 


136        OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

the  whole  four  categories  for  the  triangles,  with  the 
corresponding  averages  for  the  squares.  I  need  not 
reproduce  the  figures,  but  examination  of  Tables  11 
to  17,  inclusive,^  shows,  with  scarcely  an  exception, 
that  the  average  judgments  of  triangles  (in  column 
headed  ''T")  are  shorter  than  those  for  squares  (column 
headed  "S"). 

Next  I  will  ask  you  to  compare  the  distance-judgments 
of  similar  figures  which  contain  different  numbers  of 
pins.  According  to  our  doctrine  of  Average  Distances, 
it  should  hold  that,  of  two  triangles  of  equal  length 
of  sides,  one  arranged  with  III  pins  (a  pin  in  each 
corner)  and  the  other  with  IV  pins  (one  pin  being  in  the 
middle),  the  judgments  of  the  length  of  the  sides  of 
the  latter  should  be  shorter  than  those  of  the  former. 
In  the  second  arrangement  the  infiuence  of  the  three 
short  distances,  from  the  middle  pin  to  the  corner  pins, 
would  be  brought  into  action.  Accordingly,^  in  the 
blocks  of  Experiment  B,  with  the  increase  in  the 
number  of  pins,  should  go  shorter  judgments  from  left 
to  right  through  each  of  the  four  categories  for  triangles 
and  for  squares.  The  "  IV-pin  "  triangles  should  seem 
shorter  than  the  "  Ill-pin  "  ones  ;  those  of  "  VI  pins  " 
still  shorter  ;  and  "  VII  pins "  shorter  yet.  So  simi- 
larly for  the  IV,  V,  VIII  and  IX-pin  squares.  Examining, 
to  test  this  matter,  with  scarcely  an  exception,  throughout 

*     1  Blocks  143,  163,  188,  205,  212,  222. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.        137 

the  regular  Tables  11,  12,  13  and  11  ^  will  be  found 
precisely  the  results  that  our  above  discussion  demands. 
The  test  is  seen  best,  of  course,  in  the  footings  —  in 
their  constant  diminution  through  the  first  four  columns, 
and  again  through  the  second  four  columns. 

If  we  find  the  above  rule  less  observed  in  the  irregular 
"(a)  method  (Table  15,  Block  212),  the  reason  is  not  far 
off.  By  Law  Three,  the  closer  the  pins  the  less  would 
be  the  disassociation,  and  the  greater  the  consequent 
over-estimation  from  uncertainty."  This  would  work 
in  opposition  to  the  fundamental  tendencies  under  dis- 
cussion. In  very  short  distances,  with  triangles  and 
squares,  putting  in  extra  pins  might  work  to  lengthen 
the  distance-judgments,  just  as  we  saw  it  do  in  the 
uncertain  categories  of  Experiment  A.  Such  a  result 
would  be  most  likely  to  occur  with  the  method  that 
gave  the  most  uncertain  imiDressions  ;  in  our  case  with 
the  (a)  method.  Examination  of  the  1-cm.  judgments 
of  Block  212  will  show  this  actually  to  have  happened. 
Thus,  Avhile  the  bulk  of  the  ''regular"  B  experiments 
confirm  the  law  of  average  tendencies,  the  apparent 
exceptions  in  Table  15,  when  understood,  make  the 
confirmation  all  the  more  striking. 

§  51.  Were  the  principle  of  Average  Distances  alone 
considered,  the  sides  of  a  "  Ill-pin  "  triangle  in  Experi- 
ment B;  should  seem   precisely  equal   in  length  to  the 

1  Blocks  143,  16.3,  188,  205. 


138       OUR    NOTIONS    OF    NUIVLBER    AND    SPACE. 

same  distance  measured  between  II  pins,  as  in  Experi- 
ment A,  and  the  sides  of  the  IV  and  V-pin  squares  in  B 
shouhl  seem  longer  than  corresponding  distances  in  A. 
As  a  fact,  in  our  experiments,  all  the  judgments,  both 
for  triangles  and  for  squares,  are  shorter  in  B  than  for 
corresponding  distances  in  A.  For  this  there  are  several 
interesting  reasons. 

When,  in  Experiment  B,  we  press  the  "  III  pins  "  upon 
the  skin,  we  think  of  a  triangle,  and  not  alone  of  three 
points ;  and  to  think  of  a  triangle  is  to  think  of  its 
lineal  sides.  In  judging  and  measuring  the  pin-triangles, 
therefore,  the  thoughts  are  much  of  the  same  nature  as 
when  judging  the  lineal  figures  of  Experiment  C,  and 
it  may  be  partially  for  the  same  reasons  according  to 
which  these  last  Avere  judged  shorter  than  equal 
distances  in  Experiment  A,  that  the  triangles  and 
squares  of  B  also  seem  shorter  than  the  A  distances. 

It  may  be,  also,  that  the  simultaneous  repetition  of 
the  same  distance  in  the  three  sides  of  the  triangle,  or 
in  the  four  sides  of  the  square,  strengthens  the  spe- 
cific right-line  tendency,  and  thus  adds  foreshortening 
opposition  to  the  forces  of  uncertainty,  Avhich,  as  we 
have  found  abundantly  demonstrated,  drift  toward  over- 
estimation. 

But  the  chief  factor  in  foreshortening  our  judgments 
of  triangles  is  to  be  found,  I  suspect,  in  the  tendencies 
of  our  attention.     When  we  visually  measure  a  triangle 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.        139 

we  direct  oixr  attention  particnlarly  to  some  one  side. 
Yet  we  do  not  wholly  shut  out  of  vision  the  two  other 
sides.  The  presence  of  these  in  active  stimulation 
continually  tends  to  pull  the  focus  of  attention  towards 
themselves.  This  throws  into  the  "sum  of  tendencies" 
a  lot  of  influence  corresponding  to  what  would  actually 
happen,  did  the  eyes  sweep  narrowingly  down  the  two 
lines  towards  the  apex,  namely,  a  lot  of  shortening 
tendencies.  It  is  easily  to  be  understood,  under  our 
hypothesis,  how  this  affects  the  resultant  visual-judg- 
ment. The  illusions  discussed  by  Zollner,  Brentano 
and  others,  of  which  the  following  cut  is  a  familiar 
example,  I  should  explain  as  allied  phenomena : 


/       \ 

\       / 

To  the  extent  that  one  visualizes  in  estimating  dermal 
triangles,  will  liis  judgments  be  shortened  in  the  above- 
described  manner.  If  one  does  not  visualize,  he  brings 
in  some  other  auxiliary  mode  of  conceiving  a  triangle, 
involving  some  similar  principle.  Perhaps,  for  instance, 
that  of  drawing,  respectively,  the  thumb  and  finger 
simultaneously  down  the  sides  of  a  triangle  to  the  apex, 
and  measuring  the  parallel  distances  between  them,  in 


140        OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

terms  of  the  muscular  sensations  of  gradually  closing 
the  fingers.  In  any  case  there  is  a  narrowing  of  the 
distance-element  in  the  thought  processes,  which  makes 
itself  felt  even  when  we  use  these  processes  in  an 
auxiliary  manner  in  judging  dermal  triangles.  There 
are  abundant  reasons,  under  our  hypothesis,  therefore, 
Avhy  both  the  triangles  and  the  squares  of  Experiment 
B  seem  shorter  than  the  lines  of  Experiment  A. 

§  52.  Finally  for  this  part  of  our  paper,  having 
made  our  general  analysis  of  the  formation  of  our 
distance-judgments  from  triangles,  squares  and  circles, 
I  think  we  cannot  better  fix  the  conclusions  which  we 
liave  reached,  than  by  a  brief  examination  of  our  old 
Laws  Two  and  Three  within  these  complex  formations. 
Without  exception,  throughout  the  whole  set  of  tables 
for  Experiments  B,  C  and  D,  inclusive,  correction  of 
Law  One  leaves  an  Average  Error  with  a  +  sign.  This 
demonstrates  that  the  same  over-estimation  which 
occurred  in  Experiment  A,  holds  over  and  manifests 
itself  in  the  more  complicated  "two-dimensioned" 
judgments,  made  of  similar  distances,  and  from  the 
same  regions  of  the  body  as  formerly.  jNEoreover,  the 
greater  -f-  errors  are  shown  in  the  shorter  distance 
categories.  This  demonstrates  that  the  principle,  accord- 
ing to  which  the  over-estimation  is  distributed  in  simple 
judgments,  is  carried  over  to  express  itself  with  integ- 
rity in  those  complicated  judgments  which  are  largely 


OUR    NOTIONS    OF    NUJNIBEli   AND    SPACE.       141 

founded  upon  the  simpler  ones,  namely,  the  principle 
that  the  greatest  over-estimation  occurs  where  there  is 
the  greatest  uncertainty  and  by  reason  of  the  uncer- 
tainty.^ The  integrity  of  Law  Three  in  these  higher 
judgments  is,  therefore,  demonstrated  throughout. 

Again  throughout  all  these  tables,  without  exception, 
the  Average  Errors  of  the  longer  distances  are  minus. 
This  shows  that  the  over-estimation,  due  to  uncertainty, 
diminishes,  and  that  the  accuracy  of  the  judgments 
increases  throughout  as  the  distances  lengthen.  Thus 
is  demonstrated  that  the  influences  of  Law  Two  also 
are  carried  over  to  act  in  the  same  manner  and  with  the 
same  integrity  in  the  complicated  judgments,  whose 
formation  they  participate  in,  as  they  did  in  the  elemen- 
tary judgments,  where  we  first  discussed  them. 

§  53.  In  taking  leave  of  these  subjects  of  Xumber 
and  Distance  in  this  section  of  our  work,  we  may, 
therefore,  summarize  as  follows:  In  our  simpler  experi- 
ments we  observed  certain  principles  and  laAvs,  wliich 

1  Marked  confirmation  of  Law  Ttiree  in  the  more  complicated 
judgments  may  be  noted  by  comparing  the  results  of  Experiment 
D  with  those  of  Experiment  C.  The  apparatus  of  D  is  duller 
than  that  of  C.  The  impressions  are  more  uncertain.  From  this 
uncertainty  rises  greater  over-estimation,  which  manifests  itself 
proportionally  throughout  all  the  distances  of  D.  Were  the 
apparatus  of  the  two  experiments  equal  in  sharpness,  we  shoidd 
probably  find  the  judgments  of  solid  figures  shorter  than  of  lineal 
figures.  It  would  be  interesting  to  try  figures  formed  of  pins  set 
solidly  together. 


142       OUU    NOTIONS    OF    NUMBER    AND    SPACE. 

we  traced  out  undeviatingly  in  our  more  complicated 
experiments.  The  whole  mass  of  phenomena  we  find 
conforming  to,  and  to  that  degree  explained  by  a 
comparatively  simple,  general  hypothesis.  We  feel 
compelled,  therefore,  to  admit  this  series  of  investiga- 
tions as  peculiarly  abundant  and  convincing  evidence 
in  favor  of  the  truth  of  this  hypothesis. 


JUDGMENTS    OF    FIGURE. 

§  54.  All  the  "tendencies"  of  Number  and  all  those 
of  Distance  are  likely  to  enter  into  the  construction  of 
figure-judgments,  and  perhaps  many  more.  That  is,  we 
can  get  a  cue  for  the  judgment  that  a  certain  figure 
in  Experiment  B  is  a  triangle,  from  the  fact  that  we 
distinctly  feel  th^'ee  pins  and  only  three.  Or,  we  can 
suspect  a  lineal  figure  in  Experiment  C  to  be  a  square, 
from  a  sense  of  "distance"  fullness  "all  the  way^'ound" 
the  perimeter  of  the  otherwise  indistinguishable  im- 
pression— a  specific  feeling  which  at  once,  to  the  exj^ert, 
distinguishes  a  square  from  a  triangle  or  from  a  circle. 
Or,  we  may  reason  out  that  one  of  the  Experiment  D 
figures  is  a  circle  from  the  fact  that  no  corner  is  felt  in 
the  impression.  By  reason  of  these  several  modes  of 
getting  at  the  shape  of  any  conventional  figure,  we  are 
usually  better  able  to  tell  that  a  triangle,  square,  or 
circle  is  a  triangle,  square,  or  circle  than  to  estimate 


OUR    NOTIONS    OF    NUMBER   AND    SPACE.       143 

its  dimension,  or  count  the  exact  number  of  its  pins  or 
corners.  In  conformity  with  all  this,  it  is  now  to  be 
noted  that,  throughout  all  the  Tables  11  to  23,  the 
number  of  correct  judgments  of  figures  everywhere 
exceeds  the  number  of  correct  judgments,  either  of 
number  or  of  distance. 

By  comparing  the  distribution,  however,  of  the  ''fig- 
ure-judgments" with  that  of  the  number-judgments  in 
Experiment  B,  and  with  that  of  tlie  distance-judgments 
in  Experiments  B,  C  and  D,  they  will  all  appear  so 
similar,  at  a  glance,  as  to  convince  any  one  that  the 
same  laws  which  operated  in  the  tables  which  we  have 
already  studied,  are  also  felt  in  these  figure-judgments. 
The  general  integrity  of  our  old  Laws  Two,  Three  and 
Four,  within  this  new  field  of  mental  processes,  we 
may,  therefore,  accept  without  further  comment. 

§  55.  Table  23  demands  a  few  words  of  special 
comment.  It  is  made  up  from  the  averages  of  Tables 
18  to  20,  and  shows  the  total  averages  of  correct 
judgments,  per  100,  for  each  sort  of  figure  and  for  each 
category  of  distance;  and,  also,  it  gives  an  account  of 
the  remaining  incorrect  judgments  of  eacli  100,  in  order 
to  show  what  sort  of  errors  were  made,  and  how  many 
of  each.  By  this,  we  see  that,  in  the  shorter  and  more 
uncertain  distance-categories,  triangles  are  more  cor- 
rectly identified  than  circles,  and  circles  than  squares ; 
the  respective  averages,  for  1  cm.,  being : 


144       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


T.                          C.                          S. 

Experiment   C 

Experiment  D 

65                    44.5         i         40.5 
57.5         i         45.5         1         38.5 

And,  in  tlie  longer  and  more  certain  distances,  we  see 
these  relations  almost  precisely  reversed  ;  the  respective 
averages,  for  3.5  cm.,  being : 


T. 

1 

C. 

S. 

Experiment  C 

TlTTperiTiprit  n 

39.0 
54.5 

58.0 
83.01 

74.0 

75.0 

These  results  bring  out  the  fact  that  in  the  shorter, 
uncertain  categories  we  judge  the  contour  of  the  figure 
chiefly  by  the  acuteness  of  its  angles.  The  triangle 
having  the  sharpest  angle,  we  judge  it  most  correctly ; 
the  circle  having  no  angle,  we  pick  it  out  next  easily ; 
the  square,  lying  in  contour  between  the  two,  is  more 
difficult  to  distinguish  than  either.  The  triangle  and 
the  square,  both  having  angles,  are  more  frequently 
mistaken,  the  one  for  the  other ;  while  the  obtuse  figure 
of  the  circle,  being  more  like  the  square  than  like  the 
triangle,  the  circle  and  square  are  the  pair  next  most 
likely  to  be  mistaken,  one  for  the  other.     All  of  this 

1  Evidently  too  large.     See  preceding  Block  319. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.        145 

is  not  surprising,  and  is  not  instructive  till  we  observe 
that  the  discrimination  of  tlie  acuteness  of  an  angle  is 
something  very  kin  to  numerical  discrimination,  or  that, 
at  least,  to  judge  of  the  acuteness  of  the  angular 
impressions  involves  a  very  low  development  of  the 
distance  elements,  such  as  are  formulated  under  our 
Law  Two. 

§  56.  On  the  other  hand,  this  fact  becomes  doubly 
interesting  when,  looking  again  at  our  results,  we  find 
reason  to  believe  that,  in  the  longer  and  more  certain 
distance-categories,  the  distance  elements  yield  the  chief 
basis  for  the  judgments :  (1)  By  Law  Two,  the  distance 
elements  ought  to  be  highly  developed  in  the  higher 
distance-categories.  (2)  By  the  Law  of  Average  Dis- 
tances, where  the  distance-tendencies  are  highly  devel- 
oped and  active,  there,  other  things  being  equal,  squares 
should  seem  longer  tlian  circles,  and  circles  than  tri- 
angles. (3)  In  the  higher  distances  of  our  tables,  the 
sign  of  our  errors  is  minus  ( — ).  Consequently,  by  (1), 
(2)  and  (3),  in  the  higher  distance-categories  of  our 
Table  23,  our  theories  demand  a  greater  number  of 
correct  judgments  for  squares  than  for  circles,  and  for 
circles  than  for  triangles.  This  the  figures  quoted,  two 
paragraphs  above,  show  actually  to  be  the  state  of  the 
case.  Since  theory  and  experiments  agree,  we  conclude, 
therefore,  that  the  '' distance-elements "  are  the  active 
elements  in  these  judgments,  and  that  their  presence 


146       OUE    NOTIONS    OF    NUISIBER    AND    SPACE. 

here,  and  their  absence  in  the  lower  distance-categories, 
accounts  for  the  reversal  of  the  relations  which  the 
number  of  correct  judgments,  respectively,  of  triangles, 
circles  and  squares  bear  to  one  another,  and  which  we 
observe  in  the  figures  above  quoted. 

§  57.  One  other  matter  of  peculiar  interest  to  our 
general  subject  is  afforded  by  Table  23.  Our  triangles 
are  of  smaller  area  than  our  circles  and  squares.  Small- 
ness  goes  with  dullness  and  uncertainty  of  impression. 
When  experimenting,  therefore,  in  the  mind  of  the 
subject,  the  notion  of  "triangle"  soon  gets  intimately 
associated  with  the  lack  of  clearness  which  is  common 
to  all  the  categories  of  figure  in  the  shorter  distances 
—  triangles,  circles  and  squares — all  similarly.  Another 
way  of  expressing  the  activity  of  Law  Three  in  our 
figure-judgments  is,  therefore,  by  saying  that,  on  the 
whole,  the  errors  of  uncertainty  *'run  to  triangles." 
There  is  greater  uncertainty  in  the  shorter  distances 
than  in  the  longer  ones ;  and  the  number,  respectively, 
of  squares  and  circles  that  ''run  to  triangles,"  in 
the  1-cm.  judgments,  as  against  those  in  the  3.5-cm. 
judgments,  are:  41.0  and  24.5,  against  12.5  and  5.0,  for 
Experiment  C ;  45.5  and  24.5,  against  19.0  and  5.0,  for 
Experiment  D. 

Another  expression  of  the  same  principle  is  discovered 
by  comparing  Experiment  D  with  Experiment  C.  The 
Experiment  D  impressions  being  the  duller  (cork),  more 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.        147 

judgments  should  "run  to  triangles"  here  than  in  Exper- 
iment C.  Looking  again  at  the  figures  quoted  at  the 
end  of  the  above  paragraph,  or  at  Table  23,  directly,  we 
see  it  to  be  true,  not  only  that  the  '•  run-to-triangle '' 
errors  are  more  numerous  in  the  1-cm.  judgments  than 
in  the  o.5-cm.  ones,  but,  also,  they  are  more  numerous 
in  the  D  experiments  than  in  the  C  experiments. 

Throughout  all  our  figure-judgments,  therefore,  we 
find  the  integrity  of  all  our  laws  again  confirmed,  and 
the  whole  mass  of  empirical  results  standing  in  close 
and  illuminating  agreement  with  the  several  principles 
of  our  general  genetic  thesis. 


THE  MASS,  INTENSITY  AND    TIME-ELEMENTS    OF 
DISTANCE-JUDGMENTS. 

{Experiment  E — With  a  Moving  Pencil.) 

§  58.  The  method  and  purpose  of  Experiment  E 
have  been  explained  on  page  57.  The  general  results 
may  be  briefly  told.  Upon  all  the  regions  of  skin 
investigated,  quick  strokes  seem  shorter  than  slow  ones, 
and  light  strokes  than  heavy  ones.  Quick-light  strokes 
seem  shortest  of  all,  and  slow-heavy  ones  longest  of  all. 
The  precise  results  are  given  in  Tables  24  to  27. 

It  would,  perhaps,  be  a  popular  thing  to  explain  these 
phenomena    as    matters    of    very    simple   association. 


148       OUR    NOTIONS    OF    NUIVIBER    AND    SPACE. 

Otlier  things  being  equal,  short  "  distance  series " 
occupy  a  short  time,  and  long  distances  a  long  time. 
Thus  we  come,  on  the  whole,  to  associate  a  short-timed 
movement  with  a  short  distance,  and  a  long-timed 
movement  with  a  long  distance.  Also,  it  is  perhaps 
true  that  heavy  bodies,  on  the  whole,  move  both  less  far 
than  light  ones,  and  slower  ;  and,  thus,  both  directly 
and  indirectly,  heavy  impressions  become  associated 
with  the  notion  of  "  long  distance  "  and  light  bodies 
with  that  of  "  short  distance." 

There  is  no  doubt  whatever  that  such  associations 
can  be  and  frequently  are  formed,  and  no  doubt  that, 
through  reasoning  processes  based  upon  such  associations, 
we  frequently  arrive  at  misconceptions.  But  I  am  not  at 
all  sure  that  such  m.\s,perceptions,  such  as  our  experiments 
deal  with,  are  rightfully  explained  in  this  way.  Such  a 
theory,  when  closely  examined,  seems  to  demand  first,  a 
perception  or  sensation,  having  a  nature  in  and  of  itself ; 
then  second,  quite  another  associated  idea,  awakened 
afterward  by  this  perception  or  sensation  ;  and  then  a 
third,  still  more  subsequent  state  or  judgment,  formed 
from  the  mutual  reaction  of  the  first  and  second 
described  processes  ;  and  this  seems  to  me  a  clumsy 
affair,  involving  both  more  dogmatic  assumption  and 
more  psychological  speculation  than  is  necessary.  Why 
not  say  that  the  first  ''  whole  "  mental  state  that  rises 
to  any  sort  of  definiteness,  and  which  is  the  first  and 


OUR   NOTIONS    OF    NUMBER    AND    SPACE.       149 

native  reaction  to  tlie  outer  impressions,  as  a  whole,  has 
itself,  and  at  once,  a  specific  nature,  which  chiefly 
explains  our  phenomena  in  question  ? 

Of  course,  all  perceptions  involve  memory  processes 
more  or  less,  and  in  turn  we  must  think  of  these  as 
involving  lapse  of  time.  But  it  is  one  thing  to  talk 
about  Association  of  Ideas,  and  quite  another  to  talk 
about  Association  of  Tendencies.  It  must  be  clear 
that,  according  to  our  thesis  from  the  first,  in  us  every 
differentiated  mental  state,  however  simple,  must  be  the 
specific  correspondent  of  a  specific  Sum  or  Association 
of  Tendencies.  Even  if  we  could  isolate  and  stimulate 
any  one  single  peripheral  fibre,  it  is  probable  that  the 
cerebral  tract  Avhich  would  immediately  react  thereto, 
would  be  one  that  commonly  reacted  to  more  than  that 
one  fibre,  and  whose  habits  of  reaction  had  been  partly 
moulded  and  developed  by  the  activities  of  other  fibres. 
Its  reaction,  therefore,  even  to  the  single  fibre,  would, 
by  our  thesis,  involve  a  "  sum  of  tendencies  "  dependent 
upon  fibres  and  forms  of  outer  impressions  other  than 
the  single  one  isolated  by  our  proposition.  If  such 
"  Sums  "  are  what  we  are  to  mean  by  association,  very 
good  ;  but  they  are  very  different  matter  from  the  asso- 
ciation of  our  text-books  and  lead  to  very  different 
psychological  explanations. 

Asserting  that  a  quick-light  stimulation  of  a  definite 
stretch  of  skin  does  give  us  a  mental  reaction  which  at 


150       OUR    NOTIONS    OF   NUMBER    AND    SPACE. 

once  or  as  soon  as  it  culminates  to  any  definiteness  has 
a  specific  nature  which  is  a  perception  of  a  definite 
short  distance/  and  similarly  that  a  slow-heavy  stimu- 
lation of  the  same  stretch  gives  at  once  a  perception  of 
definite  longer  distance,  we  have  now  to  inquire  how 
our  general  thesis  would  account  for  these  phenomena. 
§  59.  Whatever  cortical  part  it  is  that  responds  to 
peripheral  stimulation,  when  it  so  responds  expression 
is  given  to  the  result  of  two  quite  independent  sources 
of  influence  —  the  one  central  and  the  other  peripheral. 
It  is  of  the  very  essence  of  our  thesis  that  the  central 
influences  or  tendencies  of  distance  are  serial  ten- 
dencies. The  peripheral  influences  of  our  experiments 
in  hand  are  also  serial.  But  what  we  may  chiefly  note 
is  that  the  two  influences  are  likely  to  tend  and  usually 
do  tend  to  different  series.  The  central  influence  is  an 
already  develo])ed  habit.  The  peripheral  influence  is 
whatever  the  experimenter  makes  it.  Theoretically,  if 
the  peripheral  influence  acted  alone,  a  specific  mental 
result  Avould  rise  in  consciousness  correspondent  to  and 

1  Of  course,  such  a  perception  does  not  come  to  us  all  named 
and  recognized,  we  will  say,  as  "  one  centimetre  "  from  the  outset. 
That  is  done  subsequently.  Yet,  it  never  would  call  up  the  proper 
recognition  and  name,  did  it  not  have  a  specific  nature  of  its  own 
to  start  with.  And  what  I  claim  is,  that  this  first-blush  has  a 
specific  distance  nature  from  the  start,  and  of  such  complicated 
development,  central  and  peripheral,  that  it  should  never  be  looked 
upon  as  a  simple  sensation,  but  as  constituting  the  very  funda- 
mental nature  of  all  distance  perception. 


OUR   NOTION'S    OF    NUMBEE,    AND    SPACE.       151 

expressive  of  a  distance  series  of  definite  length.  If 
the  central  influence  acted  alone,  a  result  expressive  of 
a  distance  series  of  different  length  would  rise.  Acting 
jointly,  as  under  outward  stimulation  they  ahvays  inust, 
the  two  influences  give  rise  to  a  distance  perception 
expressive  of  a  distance  series  mediate  between  the  two 
theoretical  ones.^  That  a  quicker  or  shorter-timed 
peripheral  movement  mediates  a  perception  of  shorter 
distance,  and  a  slower  or  longer-timed  movement 
mediates  a  longer  perception,  is  in  such  close  harmony 
with  our  theory,  as  to  the  orifjln  of  our  central  distance 
tendencies  in  general,  as  to  need  no  special  discussion. 
The  eifects  of  quick-slow  stimulation  in  Experiment  E 
are  seen,  therefore,  to  accord  with  our  general  thesis, 
and  to  be  explainable  thereby. 

§  60.  The  effects  of  liglit-heavy  stimulation  are  to 
be  accounted  for  quite  differently.-  It  has  been  the 
essence  of  our  general  thesis  from  the  start,  that  mental 

1  I  do  not  mean  to  tie  the  matter  up  to  a  hard-and-fast  mathe- 
matical relation  between  the  passing  series  and  the  average  of  all 
the  series  of  our  experience  alone.  Lots  of  other  thmgs,  as 
nutrition,  biologic  growth,  general  health,  above  all,  the  nature  of 
the  total  content  of  our  mind  at  the  moment  we  make  such 
judgments,  come  in  to  modify,  in  some  degree,  the  particular 
distance  tendencies  which  chiefly  dominate  the  focus  of  attention. 

2  Professor  Wundt  (Gnmdziige  d.  Phys.  Psy.3,  II.  19)  states 
the  fact  that  the  same  distances  appear  longer  under  heavy  than 
under  light  movement,  but  gives  no  exact  data,  nor  any  further 
explanation  of  the  phenomena  than  that  they  are  dependent  upon 
"many  physiological* and  psychological  conditions." 


152       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

distance  is  but  another  expression  for  mental  time-form. 
Not  till  we  have  time-form  can  Ave  have  "series,"'  and 
in  "series"  is  the  origin  of  all  mental  distance.  Right 
here  we  should  note  that  mental  processes  may  differ  in 
other  ways  than  in  time-form.  For  instance,  the  most 
elementary  feelings  may  differ  in  <'mass";  the  specific 
feeling  resulting  from  the  reaction  of  a  large  nervous 
mass  is  not  the  same  as  results  from  a  smaller  nervous 
mass.  A  big  pain  is  not  the  same  thing  as  a  little  pain, 
even  outside  of  intensity.  Now,  just  because  mental 
processes  may  differ  in  other  respects  than  in  time-form, 
and  because  mental  distance  is  solely  time-form,  there- 
fore, we  should  clearly  observe  that,  by  our  thesis, 
mental  distance  cannot  be  conditioned  by  mental  mass. 
Definitely  stated :  The  time-form  remaining  the  same, 
"more"  or  "less"  feeling  can  never  constitute  mental 
distance.  I  will  emphasize  this  by  an  illustration.  Our 
thesis  has  declared  from  the  start  that,  othei;  things 
being  the  same,  originally,  the  simnltaneous  stimula- 
tion of  any  number  of  nerve-ends  would,  mentally,  give 
the  same  "distance"  result,  whatever  the  peripheral 
arrangement  of  these  nerves  —  whether  in  a  bunch  or  in 
a  straight  line.  Not  until  successive  stimulation  came 
in,  could  the  mental  results  differ  in  "length."  Yet,  it  is 
to  be  observed,  the  results  of  simultaneous  stimulation 
could  differ  otherwise  than  in  "length";  for  the  stim- 
ulation of  many  nerves  would  give  yiore  feeling  than 


OUR    NOTIONS    OF    NUiMBER    AND    SPACE.       153 

would  the  stimulation  of  a  few.  Thus,  there  could  be 
more  feeling,  yet  not  more  '-length"  ;  more  "mass,"'  yet 
not  more  "distance." 

To  recognize  this  independence  of  mental  '''mass"  and 
mental  "distance"  is  very  important,  and  failure  to  do 
so  constitutes  a  crucial  error  in  all  psycho-physical 
experimentation  into  which  "space"  enters  in  any  way. 
The  root  of  this  error  lies  in  assuming,  outright,  that 
the  amount  of  stimulation  applied,  peripherally,  in  any 
given  unit  of  time,  necessarily  bears  a  direct  proportion 
to  the  amount,  or  "??««sa%"  of  feeUnrj  resultant  therefrom 
in  the  corresponding  unit  of  time.  Xo  doubt  it  is  true 
of  the  physical  activity  to  Avhieh  the  feeling  ultimately 
corresponds,  that  the  amount  of  its  activity  bears  a 
direct  proportion  to  the  "mass"  of  the  feeling.  But 
the  mistake  comes  in  by  assuming  that  the  amount  of 
this  nltininte  physical  activity  is  always,  for  correspond- 
ent units  of  time,  directly  proportional  to  the  amount 
of  peripheralhj  applied  stimulation.  As  well  might  we 
assume  that  mental  reaction-times  always  correspond 
directly  with  the  time-form  of  the  outer  stimulation, 
and,  therefore,  are  independent  of  the  amount  of  periph- 
eral area  stimulated — (that  the  reaction-time  to  a  small 
heated  surface  is  the  same  as  to  a  larger  heated  surface, 
the  intensity  of  stimulation  being  a  constant).  Indeed, 
to  assume,  in  Experiment  E,  that  the  resulting  distance- 
series  would  be,  in  length,  independent  of  the  heft,  or 


154       OUR    NOTIONS    OF   NUMBER    AND    SPACE. 

"amount,"  of  stimulation  per  time-unit  of  the  moving 
pencil,  would  be  identical  with  asserting  that  mental 
time-actions  are  independent  of  amount  of  peripheral 
stimulation.  In  view  of  our  general  thesis,  and  of  all 
it  has  explained  to  us  in  our  foregoing  experiments, 
until  it  can  be  shown  that  the  distance-series,  or  ten- 
dencies, do  not  change  in  length,  proportionally  to  the 
heft  of  the  pencil,  I  am  inclined  to  hold  to  the  doctrine 
that  the  differences  shown  between  the  judgments  of 
the  "light"  and  the  "heavy"  categories  of  Experiment 
E,  are,  in  so  far  as  distance  is  concerned,  wholly  differ- 
ences of  mental  time-form  and  are  not  at  all  differences 
of  "mental  mass." 

§  61.  This  being  so,  it  is  incumbent  upon  our 
hypothesis  to  show  how  the  heft  of  the  pencil  could 
affect  the  distance-series.  I  think  it  does  so  in  two 
different  ways.  First,  the  skin  and  tlie  flesh  beneath, 
being  flexible,  the  harder  the  pencil  is  pressed  the  more 
is  its  influence  spread  peripherally,  both  by  pressure 
and  by  tension,  and,  consequently,  the  larger  is  the 
stretch  of  skin  actually  stimulated.  Second,  it  is  not 
unfair  to  suppose  that  the  more  intense  any  stimula- 
tion is,  the  longer  may  its  results  continue,  centrally, 
—  continue,  either  by  penetrating  through  a  longer 
series  of  cells,  or,  by  more  prolonged  activity  of  the 
same  cells.  Thus,  we  see  how  a  part  of  the  heft  of  the 
pencil  can  be  transformed   into  "length"    of    resulting 


OUR    NOTIONS    OF    NCTMBEE,   AND    SPACE.       155 

distance-perceptions,  without  committing  the  mistake 
(which  we  have  called  the  "root  of  error"  in  much 
modern  psycho-physical  experimentation)  of  taking  for 
granted  that,  either  tlie  intensity,  or  the  mass  of  the 
resultant  feeling,  or  of  the  ultimate  central  physical 
activity,  need  to  bear  any  constant  ratio  Avhatever  to  the 
intensity  or  amount  of  the  peripheral  stimulation.^ 

I  am  inclined  to  account  for  the  over-estimation  of 
the  "  heavy "  stimulations  of  Experiment  E  by  the 
two  above  explanations,  rather  than  by  saying  that  we 
get  any  notion  of  longer  distance  by  reason  of  the 
greater  ''volume,"  or  "mass,"  or  the  greater  intensity 
of  feeling  which,  undoubtedly,  results  from  the  harder 
pressure  of  the  pencil.^ 

1  As  no  one  has,  to  my  knowledge,  ever  claimed  that  mere 
intensity  formed  a  part  of  the  mental  constituency  of  space,  I 
need  not  discuss  such  a  proposition  here.  Possibly,  intensity  is 
but  serial  relationship  between  time-form  and  mass.  If  Professor 
Charles  Pierce  is  correct,  then  the  physical  atoms,  and  all  kinetic 
phenomena,  are  but  different  time-manifestations  of  that  absolute 
continuum,  in  which  all  physical  and  psychical  things  find  ultimate 
identity. 

2  Professor  James,  in  his  theories  of  space  and  distance,  makes 
great  use  of  "The  Feeling  of  Crude  Extensity,"  .  .  .  "the  orig- 
inal sensation  out  of  which  all  the  exact  knowledge  of  space  .  .  . 
is  woven"  (James'  Psychology,  II,  134,  135).  The  doctrine  I 
advocate  holds  that  our  every  notion  of  extensity  is  wholly  an 
expression  of  time-extension.  If  "crude"  and  "original"  mean 
"independently  of  time-form,"  then  I  should  say  that,  independent 
of  time-form,  there  could  be  no  "feeling  of  extensity,"  while, 
theoretically,  there  might  still  be  "big  feelings"  and  "little  feel- 
ings," in  the  sense  of  "more  feeling"  and  "less  feeling."     Of 


156       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 


EXPERIMENT    G.  — "WITH   A   SINGLE   PIN. 

§  62.  Our  Law  Three  has  been  based  upon  the  fact 
that  when,  for  a  given  stimulation,  no  specific  habit 
has  been  sufficiently  developed  to  react,  under  all  condi- 
tions, with  exactness,  then  the   influences  of  a  lot  of 

course,  without  any  "mass,"  or  content,  at  all,  there  would  be  no 
feeling.  And  if  crude  mass,  or  content,  of  any  kind,  is  what 
Professor  James  means  by  "original  sensation,"  why,  then,  every 
mental  state  is  woven  out  of  such,  but,  as  much  so,  our  emotional 
states,  and  all  other  "states,"  as  our  knowledge  of  space.  Yet, 
I  am  inclined  to  think  that  this  man  of  genius  has, '  as  usual, 
laid  hands  on  an  important  truth.  While  I  deny  that,  '■'■all  our 
exact  knowledge  of  space"  is  woven  out  of  this  "mass "-element 
of  mental  variability,  yet  I  believe  that  a  part  of  the  difference 
between  different  space-perceptions  may  be  a  difference  in  their 
respective  mass-elements.  It  is  true  that  a  "big  sound"  may 
form  a  part  of  our  perception  of  a  big  space ;  but  it  is  also  true 
that  a  "small"  light  may  form  a  part  of  our  perception  of  a 
vastly  larger  space,  as  of  a  distant  star.  "What  I  doubt,  in  criti- 
cism of  Professor  James's  theories,  is,  that  my  spatial  perceptions 
of  a  star  are  "aZr'  woven  out  of  the  "mass"-element,of  mental 
variability,  or  even  any  important  part  of  them.  Professor  James 
says,  of  number,  that  "all  would  be  one  big,  buzzing,  blooming 
confusion,"  till  the  first  shock  of  succession  occurred,  and,  that, 
out  of  that,  or  in  it,  would  the  feeling  of  duality  arise.  Precisely 
similarly  of  distance.  I  say  there  might  be  big  buzzings  and  little 
buzzings,  big  blooms  and  little  blooms,  but  never  would  any  feel- 
ing of  distance  arise  till  some  feeling  had  buzzed  or  bloomed  so 
long  a  time.  That  buzzing  or  blooming  "so  long,"  would  be  the 
original  distance-event ;  and,  whatever  consolidated,  or  modified, 
"  memory  "-event  might,  subsequently,  rise  as  the  resultant  repre- 
sentation of  that  new  feeling,  would  be  wholly  proportional  to, 
and  expressive  of,  the  "so  long"  of  the  original  event,  and  in  no 
way  proportional  to  its  bigness  or  its  littleness. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       157 

allied  but  looser  tendencies  are  awakened  to  fill  out 
the  resulting  perception.  Hence,  "over-estimations." 
The  question  now  arises  :  What  would  happen,  if  a  lot 
of  these  allied  tendencies  should  be  awakened  by- 
other  means  than  tlirougli  the  uncertainty  of  the  outer 
stimulations  alone,  as,  for  instance,  through  strongly 
allied  inner  associations,  passing  in  the  current  of  our 
thoughts  at  the  moment  the  outer  stimulations  were 
made  ?  Do  our  passing  thoughts,  also,  modify  our  per- 
ceptions under  Law  Three  ? 

§  63.  The  following  experiment  gives  us  very  curi- 
ous information  on  this  point,  enabling  us  to  connect, 
in  an  instructive  manner,  our  foregoing  experiments 
and  thesis  regarding  outer  Perceptions,  with  the  more 
common  doctrines  regarding  inner  Associations. 

In  proper  holders,  prepare  several  sets  of  medium- 
sized  sewing-needles,  two  in  each  set,  in  such  ways  that 
the  distance  between  the  points,  in  each  set,  shall  be 
regularly  graded,  from  1  mm.  up  to  8  or  10  cm.  apart. 
Also  prepare  one  "control"  holder,  bearing  a  single 
pin  of  medium  size  and  fineness  of  j^oint.  It  is  best,  at 
the  outset,  to  show  these  needles  (not  the  "control")  to 
the  person  to  be  experimented  upon,  and  to  make  sure 
that  he  is  visually  appreciative  of  the  exceeding  minute- 
ness of  the  minimal  distance.  Until  the  experiment  is 
concluded,  the  subject  should  never  be  permitted  to 
suspect  that  there  is  any  "control"  holder,  containing 
a  single  pin. 


158       OUK    NOTIONS    OF    NUMBER    AND    SPACE. 

Choose,  now,  any  part  of  the  body,  the  tip  of  the 
tongue,  for  several  reasons,  being  preferable  to  begin 
with.  Select,  at  first,  such  a  pair  of  needles  that  their 
two  points  will,  with  certainty,  be  perceived  double 
when  applied  to  the  region  of  skin  chosen.  Continue  to 
apply  the  points,  stepping  them  about,  here  and  there  ; 
sometimes  pressing  both,  and,  again,  only  one,  till  the 
subject  is  perfectly  familiar,  not  only  Avith  the  prick 
of  the  single  points,  but  also  with  the  two  as  "felt 
together."  Then  proceed  likewise  with  the  series  of 
points,  using  each  pair  in  order,  down  toward  those  of 
the  minimal  distance  apart. 

As  soon  as  the  subject  becomes  uncertain  in  his  judg- 
ments, begin  to  explain  to  him  how  much  concentration 
and  attention  and  practice  will  do  toward  increasing  his 
power  of  discrimination.  Perhaps  it  is  well  here  to  read 
to  him,  from  Ladd's  text-book  (Phy.  Psy.,  page  411), 
that  Goldscheider  could,  on  the  finger,  feel  twQ  points 
only  2  mm.  apart,  and  actually  to  show  him  a  pair  of 
needles  with  points  separated  by  that  distance.  Then 
begin  experimenting  again,  and,  just  as  the  judgments 
grow  uncertain,  introduce  the  "control"  pin,  unknown 
to  the  subject,  and  keep  him  well  up  in  the  effort  of 
discrimination.  Having  performed  this  experiment  on 
many  persons,  I  have  never  yet  failed,  with  proper  care 
in  introducing  the  "control,"  to  procure,  in  time,  a 
series  of  successive  judgments  which,  when  once  begun, 


orn  NOTIONS  of  number  and  space.     159 

may  be  continued  indefinitely,  and  in  all  of  which,  the 
single  pin-prick  is  clearly  and  uniformly  perceived  as 
double. 

§  04.  Whatever  its  relation  to  the  outer  object  and 
to  nerve-ends  outwardly  stimulated,  the  mental  state 
itself  is  unmistakably  real  and  distinct ;  the  two  points 
are  as  sharply  felt  separate  as  ever  any  two  points  can 
be,  and  continuously  and  evenly  so,  for  so  long  as  the 
pins  remain  applied.  The  pins  may  be  held  in  the 
subject's  own  grasp,  and  rocked  back  and  forth,  causing 
now  one  point  to  be  felt,  and  now  the  other ;  and  so 
with  all  sorts  of  varying  intensity,  consequent  upon 
''feeling  around"  with  them, — all  precisely  as  if  there 
really  were  two  pins. 

§  65.  The  line  of  direction,  between  the  two  points, 
may  also  be  discriminated,  with  reference  to  the  topog- 
raphy of  the  region  of  skin  worked  upon,  and  to  sur- 
rounding objects,  as  definitely  as  any  perception  of 
direction  ever  is.  If  the  conditions  of  application 
remain  unchanged,  and  in  the  absence  of  any  conscious 
suggestion,  either  from  any  outward  circumstance,  or 
even  as  prompted  from  within  one's  own  imagination, 
then  the  seeming  line  of  (^rection  is  likely  to  remain, 
to  the  subject,  imchanged  also.  Yet  the  apparent  direc- 
tion is,  plainly,  not  one  of  passing  accident.  Xearly 
every  region  has  a  most  constant  direction,  one  strong- 
est developed  and  most  natural  to  rise  ;  a  sort  of  "local 


160       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

sign,"  dependent  upon  the  contour  and  local  expression 
of  eacli  particular  region.  As,  for  instance,  on  the  trunk 
and  limbs,  in  the  absence  of  all  suggestion  to  the  con- 
trary, the  direction  is  more  likely  to  appear  longitudinal 
than  transverse.  The  more  highly  discriminative  the 
region  is  spatially,  the  less  constant  is  the  native  direc- 
tion likely  to  be.  Upon  the  tongue,  the  direction  is 
likely  to  change  whimsically,  upon  the  slightest  move- 
ment given  to  the  pin.  The  fingers,  though  as  delicately 
sensitive  as  the  face,  are  less  constant  in  this  native 
direction.  The  slightest  passing  suggestion,  however, 
is  likely  to  establish  a  corresponding  notion  of  direction 
for  the  points.  If  the  subject,  just  as  he  closes  his 
eyes,  sees  you  raise  toward  liis  forehead  a  holder  really 
having  two  pins  so  set  that  when  applied  they  would 
naturally  fall  vertically,  and  you  then  deftly  substitute 
the  "control"  without  disturbing  his  expectations,  the 
line  of  direction  is  pretty  sure  to  be  felt  vertically. 
Thereafter,  the  direction  will  seem  to  change  by  reason 
of  the  most  unconscious  suggestions.  If  the  subject 
take  the  holder  in  his  own  hand,  and  keeping  his  elbow 
as  nearly  in  one  point  as  possible,  step  the  pin  along 
naturally  from  the  center  of  the  forehead  around  and 
down  tOAvard  the  ear,  the  direction  will  constantly  move 
to  correspond  to  the  radius  of  the  arm  in  its  several 
positions.  The  changing  of  the  experimenter's  arm  will 
also   change   the   direction   accordingly.     In   short,  the 


OUR   NOTIONS    OF    NUMBER    AND    SPACE.       161 

direction  once  established  for  the  immediately  subse- 
quent applications,  it  will  usually  appear  to  follow  all 
the  changes  which  it  ought  to  follow,  were  two  real 
pins,  rigidly  fixed  with  reference  to  each  other,  applied 
by  precisely  similar  movements. 

§  66.  More  surprising  still  is  the  apparent  distance 
between  the  two  fictitious  points.  This  is  different,  in 
the  absence  of  positive  suggestions  to  the  contrary,  for 
each  particular  region  of  the  body;  and  it  appears 
pretty  nearly  to  correspond  to  the  distance  given  by 
Weber  as  the  minimum  at  which  two  compass  points 
are  perceived  double  for  the  same  region  of  skin. 

For  instance,  the  subject  having  been  brought  to  feel 
the  "control  pin"  double  upon  the  tip  of  the  tongue,  if 
then  he  be  asked  what  the  distance  seems,  his  judgment 
is  likely  to  be  in  the  neighborhood  of  3  mm.  If,  how- 
ever, the  pin  be  stepped  slowly  along  the  side,  or  rim, 
of  the  tongue,  the  distance  will  spontaneously,  and 
without  any  suggestion  of  any  such  occurrence,  seem 
to  him  to  spread  to  four,  five,  or  perhaps  more,  milli- 
meters apart.  Upon  the  middle  upper  surface  of  the 
tongue,  the  distance  is  likely  to  seem  2  cm.  In  short, 
whenever  the  pin  is  stepped  from  place  to  place  over 
parts  having  different  native  space-values,  the  apparent 
distance  between  the  points  changes  spontaneously  in 
accordance  with  these  native  values.  And  all  this  will 
happen,  though  the  experiments  be  so  managed,  as  were 


162       OUE    NOTIONS    OF   NUAIBER    AND    SPACE. 

our  own,  that  the  subject  shall  all  the  time  be  under  the 
impression  that  compass  points  have  been  set  at  varying 
distances,  which  he  is  to  estimate  in  order  to  test  the 
accuracy  of  his  judgments.  This  "spreading"  phenom- 
enon is  particularly  marked  upon  the  forehead,  where, 
throughout,  the  "native  distances"  are  pronouncedly 
irregular.  In  the  centre  of  the  forehead,  the  distance 
is  likely  to  seem  one  or  one  and  a  half  centimeters ; 
stepped  along  horizontally  to  a  point  above  the  tip  of 
the  ear,  it  will  spread  till  it  appears  like  three  or  four 
centimeters. 

That  there  is  a  natively  developed  constant-distance 
which,  in  the  absence  of  all  passing  suggestion,  spon- 
taneously rises  to  shove  itself  in  between  the  two 
points,  when,  by  proper  management,  these  have  been 
psychologically  separated  for  the  subject ;  and  that  this 
constant  is  definite  for  each  particular  region  of  skin, 
but  different  for  different  regions  of  skin,  is  a  fact 
which  almost  any  one,  by  proper  care,  can  with  great 
ease  demonstrate  to  himself.  From  the  results  of  a 
tolerably  large  number  of  tests  upon  four  persons,  and 
of  less  numerous  tests  upon  ten  other  persons,  I  got  the 
following  averages  for  these  "native  distance-constants" 
of  the  regions  specified.  I  must  state,  however,  that 
the  average  error  for  these  figures  was  considerable, 
and,  altogether,  I  have  not  had  the  opportunity,  as  yet, 
to  determine  the  matter  for  the  several  regions  of  the 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       163 

body  as  fully  as  the  importance  and  significance  of  the 
phenomenon  seems  to  promise  would  be  profitable  : 

TABLE    OF   NATURAL   DISTANCES. 


Tip    of  tongue   . 

Side  of  tongue   . 

Top   of  tongue,  middle 

Rim  of  lower  lip,  middle 

.3  mi 

5     " 

10     " 

4     " 

Rim  of  lower  lip,  side 

5     " 

Forehead,  middle 

10     " 

Forehead,  side    . 

20     " 

Scalp,  over  ear  . 
Ball  of  forefinger 

30     " 
4     " 

Back  of  hand     . 

10  to  15     " 

Forearm 

15  to  25     " 

Abdomen    . 

35 

to  50     " 

§  67.  According  to  the  above  method  of  procedure, 
the  subject  is  to  be  kept  in  entire  ignorance  that  a 
single,  or  "control"  pin  is  ever  being  used  upon  him. 
This,  I  think,  is  advisable,  if  not  absolutely  necessary, 
for  every  first  demonstration  of  the  experiment  upon  a 
novice.  But  after  a  person  has  once  been  operated 
upon  in  this  manner,  and  has  himself  experienced  how 
perfectly  real  and  distinct  the  illusions  are,  he  may 
then  successfully  operate  upon  himself  with  equal  satis- 
faction. The  vividness  of  the  results  improve-  rapidly 
with  practice,  until  it  becomes  quite  possible  for  one  to 
carry  the  phenomenon  in  an  unbroken  series  from  the 
lips  to  the  finger  tips,  and  thence,  backward  and  down- 
ward across  the  abdomen  and  thighs,  to  the  points  of 


164       OUU    NOTIONS    OF    NUMBER    AND    SPACE. 

the  toes,  the  distance  between  the  pins  seeming  to  vary 
the  while  from  a  few  millimeters  to  as  many  centi- 
meters, according  to  the  native  distance-value  of  the 
region  traversed. 

And  what  we  particularly  call  attention  to  is,  that 
the  distance  apart  at  which  the  subject  begins  to  be 
uncertain  of  liis  ability  to  perceive  the  pins  sepa- 
rately, is  approximately  the  distance  which,  for  the 
same  region,  they  appear  apart  when  fictitiously  pre- 
sented by  the  "control"  pin.  Another  way  of  stating 
this  is,  that  Weber's  distances  are  approximately  our 
native  distances.  We  shall,  however,  soon  come  to  look 
upon  Weber's  distances  as  less  constant  functions,  and 
of  far  different  significance,  than  has  been  commonly 
conceived  in  the  abundant  literature  devoted  to  them. 

Not  only  may  two  fictitious  ])oints  be  perceived  from 
the  single  pin,  but  upon  the  tongue  and  lips  at  least 
three,  four  and  even  five  can  be  felt.  And  these  may 
be  felt  as  if  arranged  somtimes  in  lines,  sometimes  in 
triangles,  and  sometimes  in  parallelograms  or  squares. 

Such  being  the  phenomena  and  the  method  of  their 
experimentation,  we  may  now  seek  their  explanation 
and  their  significance  under  our  general  thesis. 

§  68.  Every  perception  is  the  resultant  of  a  sum  of 
tendencies,  partly  peripheral,  and  partly  associative  or 
central.  In  ordinary  perception,  we  are  apt  to  underrate 
both  the  amount  and  the  character  of  the  portion  filled 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       165 

in  by  the  central  influences.  Before  me  now  is  a  curi- 
ous picture.  When  I  look  at  it  expecting  to  see  a  certain 
face  of  an  old  woman,  I  see  it  as  vividly  as  in  an  ordi- 
nary portrait.  When  I  look  expecting  a  certain  girlish 
face,  I  see  that  equally  well.  The  image  thrown  on  the 
retina  is  the  same  in  the  two  cases.  The  portions  filled 
in  centrally  are  very  different.  In  proportion  to  their 
difference,  we  must  appreciate  how  great  was  the 
amount  of  the  central  contribution  to  each  perception. 
Also,  realizing  how  all  parts  of  the  perception  appear 
equally  vivid,  objective  and  real,  we  are  reminded  that 
there  is  no  difference  to  be  distinguished  in  kind  or 
quality  between  the  portions  which  are  due  directly  to 
peripheral  excitation  and  those  which  are  filled  in 
through  associative,  or  central  excitation. 

What  is  requisite  in  such  cases  of  two  possible  per- 
ceptions for  the  same  outward  stimulation,  seems  to  be, 
that  they  both  shall  be  of  such  a  nature  that  their 
peripheral  elements  are  identical.  A  certain  drawing 
may  be  perceived  as  two  straight  lines,  or  as  a  cross,  but 
never  as  a  circle.  The  peripheral  elements  for  the  first 
two  of  these  perceptions  would  be  identical;  they  would 
not  be  identical  for  the  last  two. 

When  a  pin  is  pressed  upon  the  flexible  skin  ever  so 
lightly,  a  group  of  several  nerve-fibers  is  pretty  sure 
to  receive  stimulation.  What  sort  of  perception  will 
follow,  will  not  depend  upon  the  number  of  nervjsrefids, 


166       OUR   NOTIONS    OF    NUMBER   AND    SPACE. 

nor  upon  their  peripheral  grouping — ^  whether  in  one 
bunch  or  in  two — but  upon  the  portion  filled  in  or  not, 
or  centrally.  The  '^feeling  of  double"  originates  from 
a  shock  of  succession.  But  once  originated,  it  may  be 
revived  and  filled  in  centrally,  precisely  like  any  other 
centrally  contributed  portion  of  any  perception.  If  the 
same  set  of  nerves  from  whose  successive  stimulation  a 
"feeling  of  double"  originated,  is  thereafter  simultane- 
ously stimulated,  a  perception  of  two  points  may  result 
therefrom.  Yet,  we  have  to  note  that  such  will  not 
necessarily  result.  What  we  have  thereafter  is  the 
possibility  of  two  different  perceptions  from  the  same 
peripheral  stimulation.  Apparently,  the  peripheral  ele- 
ments are  the  same  in  both.  If  the  definite  central 
influence  acts  jointly  with  the  definite  peripheral  influ- 
ence, we  perceive  two  points.  If  the  peripheral 
influence  acts  alone,  we  perceive  one  point.  The  ques- 
tion of  perceiving  two  points  or  one  is,  therefpre,  the 
question  of  the  proper  central  influence  being  or  not 
being  brought  into  activity  conjointly  with  the  proper 
peripheral  influence,  and  without  regard  to  how  it  shall 
be  awakened. 

In  our  experiment  with  the  single  pin,  it  is  not  diffi- 
cult to  see  how  the  proper  central  influence  is  awakened. 
We  are  expecting  to  feel  two  points  at  precisely  the 
place  touched  by  the  single  pin.  If  the  same  central 
parts  are    active   in   expectation  which   are    active   in 


OUR    NOTIONS    OF    NU:MBER    AND    SPACE.       167 

realization,  —  and,  from  modern  psychology,  it  is  fair  to 
suppose  they  are,  —  then  we  may  say  that  all  the  central 
influences  requisite  to  a  perception  of  two  points  at  the 
place  in  question,  are  already  in  partial  activity  through 
the  expectation.  Consequently,  when  the  proper  periph- 
eral influence  is  aroused  by  the  pressure  of  the  pin,  all 
the  conditions  requisite  to  the  perception  of  two  pins 
are  fulfilled.  The  proper  sensational  tang  and  strength 
of  feeling  is  furnished  through  the  peripheral  influence, 
and  the  doubleness  is  achieved  through  the  filling  in  of 
the  proper  central  influence.  The  whole  thing  is  possi- 
ble, through  the  congruity  of  the  peripheral  elements 
furnished  by  the  passing  outward  stimulation  with  the 
peripheral  elements  demanded  by  the  expected  percep- 
tion. Were  they  not  alike,  they  would  either  inhibit 
each  other  or  compromise,  by  calling  up  some  resultant 
perception  different  from  either. 

In  short,  therefore,  the  reason  that  we  do  not  perceive 
a  pin-point  double  is,  because  the  group  of  nerves  so 
stimulated  has  not  of  itself  sufficient  associative  con- 
nection with  the  proper  central  influence  to  call  it  up 
jointly.  The  reason  we  do  perceive  the  pin  double  in 
our  experiment  is,  because  the  expectation  provides  the 
proper  central  influence.  The  reasons  we  cannot  double 
up  any  and  every  sort  of  perception,  by  expecting  to  per- 
ceive it  double  under  outward  conditions  that  commonly 
present  it  to  us  single,  are  of  two  kinds.     First,  if  the 


168       OUE,   NOTIONS    OF    NUMBER    AND    SPACE. 

perception  is  complicated,  the  peripheral  elements  actu- 
ally furnished  from  without,  are  the  less  likely  to  agree 
with  the  elements  furnished  from  within  the  expecta- 
tion ;  i.e.,  the  more  difficult  is  it  for  the  expectation 
and  the  realization  to  agree  in  those  elements  wherein 
they  must  agree.  And,  second,  the  more  complicated 
the  perception,  the  more  difficult  is  it  for  us  to  expect 
it  to  be  doubled.  If  only  one  operator  were  present, 
it  would  be  impossible  for  the  subject  to  expect  two 
right  hands  to  be  laid  on  his  breast  at  once.  Only  very 
simple  perceptions,  similar  to  those  which  frequently 
are  perceived  double,  could  ever  be  successfully  manip- 
ulated by  the  method  of  our  present  experiment. 

§  69.  The  distance  shoved  in  between  each  pair  of  our 
fictitious  points  is  now  easily  to  be  explained.  Having 
explained  how,  through  the  expectation  of  the  subject, 
the  proper  central  influence  Avas  aroused  requisite  to  the 
perception  of  doubleness,  I  need  only  to  say,  naw,  that 
the  fact  that  some  sort  of  distance  was  also  perceived 
under  the  same  circumstances  is  to  be  explained  in  a 
similar  way.  It  remains  to  make  plain  how  it  happens 
that  a  certain  definite  lengtli  of  distance  is  called  up 
constantly  for  each  particular  region,  and  a  different 
length  for  different  regions. 

This,  after  our  many  discussions,  is  not  difficult. 
Every  point  of  our  skin  has  during  life  been  knit  up 
into  many  serial  stimulations,  and,  proportionally,  has 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       169 

become  joined  to  as  many  different  distance-tendencies. 
If  the  habits  of  reaction  have  not  been  narrowed  to 
some  particular  length,  then,  upon  stimulation,  come  in 
all  the  results  of  uncertainty,  so  abundantly  formulated 
by  us  under  Law  Three.  The  distance  phenomena  of 
ouf  Experiment  G  are  but  extreme  instances  of  such 
uncertainty.  The  subject  expects  some  sort  of  dis- 
tance. He  is  not  precisely  sure  how  the  needles,  or  the 
compass  points,  will  be  set.  Upon  application  of  the 
"control"  pin,  his  mind  ranges  up  and  down  Avithin 
the  limits  of  his  expectation,  precisely  as  upon  the 
application  of  any  piece  of  apparatus  from  Experiment 
A.  Presently,  as  the  resultant  of  the  sum  of  all  the 
tendencies  so  awakened,  both  from  the  pin  and  from 
the  expecting,  a  definite  perception  rises,  precisely  as 
in  any  other  judgment  of  distance.  The  enly  things 
remarkable  about  such  occurrences  are  the  weakness 
and  uncertainty  of  the  peripheral  tendencies  awakened 
by  the  single  pin,  and  the  unusual  strength  of  the 
central  tendencies  awakened  by  the  intensity  of  the 
expectation.  To  some,  it  may  not  yet  seem  quite  clear 
why  the  fictitious  distance  should  be  so  different  for 
different  regions.^     They  vary  for  precisely  the  reason 

^  If  our  Experiment  G  distances  vary  considerably  from  those 
given  by  Weber,  I  am  inclined  to  think  this  is  chiefly  due  to  the 
fact  that  we  used  pin-points,  and  he  used  rounded  points  of  a 
millimeter  in  diameter. 


170       OUR    NOTIONS    OF    NUjSIBER    AND    SPACE. 

that  Weber's  distances  vary,  namely,  because,  througli 
life,  the  distance  experiences  of  the  different  regions 
have  been  different.  In  turn,  this,  j)artly,  is  due  to 
gross  anatomy,  and  partly  to  the  exigencies  of  life.  A 
long  line  can  be  drawn  on  the  leg ;  not  on  the  tongue. 
§  70.  It  should  be  plain  now,  I  think,  why,  when  we 
expect  the  proper  distance  in  our  G  experiment,  we 
perceive  it  fictitiously.  It  only  remains  to  make  clear 
why,  when  we  expect  any  other  distance  than  the  proper 
one,  the  ''control"  pin  will  give  us  no  perception  of 
distance  whatever.^  For  instance  :  Why,  on  the  abdo- 
men, if  we  expect  two  points,  3  mm.  apart,  do  we 
get  no  perception  ?  Or,  why,  on  the  arm,  if  we 
expect  3  cm.,  do  we  get  nothing?  To  answer  this: 
we  think  in  various  terms,  sometimes  in  visual  images, 
sometimes  in  muscular  images.  No  doubt,  dermal 
images  enter  into  all  of  our  "certain"  judgments  in 
the  foregoing  experiments.  All  we  have  learnecj  about 
development  of  central  influences  and  tendencies 
through  peripheral  experiences  implies  this.  ''Uncer- 
tainty" implies  imperfect  development  of  dermal 
images.  We  can  visualize  a  nine-pin  square  resting  on 
our  arm  at  any  time  with  accuracy.  We  can  perceive 
it  dermally  with  uncertainty ;  and  I  am  inclined  to 
believe  that  we  are. less  able  to  expect  it  in  any  given 

1  Of  course,  I  speak  approximately.     Both  the  proper  distance 
and  the  expected  distance  have  a  short,  permissible  range. 


OFR   NOTIONS    OF    NUMBER   AND    SPACE.       171 

place,  in  dermal  images,  than  we  are  to  perceive  it,  in 
terms  of  dermal  feeling,  in  the  same  place. 

This  being  so,  I  think  the  answer  to  our  above 
questions  is,  that  when  we  expect  any  sub-threshold 
distance  for  any  region  of  skin,  we  do  not  expect  it  in 
dermal  images,  but  only  in  visual  or  muscular  im- 
ages. Consequently,  no  dermal  tendencies  are  aroused, 
such  as  are  requisite  to  cooperate  with  the  peripheral 
stimulation  to  produce  a  dermal  perception.  Indeed, 
I  suspect  that  the  reason  we  cannot  discriminate 
the  sub- Weber  distances  is  just  because,  on  account 
of  their  not  being  sufficiently  developed,  we  cannot 
expect  them  or  conceive  them  in  proper  dermal  terms. 
When  we  expect  the  proper  distances,  we  can  conceive 
this  in  dermal  terms,  and,  therefore,  the  perception  is 
possible.  When  we  expect  too  long  a  distance,  here, 
also,  the  dermal  images  are  lacking ;  for  the  range  of 
distance-tendencies,  developed  with  any  sort  of  perma- 
nent central  efficiency  for  any  given  limited  area  of  skin, 
is,  I  suspect,  very  limited.  What  we  get  in  our  ficti- 
tious G  perceptions  seems  to  be  a  general  average,  or 
resultant,  of  this  rather  narrow  range  of  habitual  dis- 
tance-reactions for  tlie  given  very  limited  area  on  and 
immediately  around  which  the  pin  is  pressed,  or  ex- 
pected to  be  pressed.  And,  of  course,  this  is  constant 
for  a  particular  region  and  different  for  different 
regions. 


172       OUR   NOTIONS    OF    NURIBER   AND    SPACE. 

EDUCATION     OF     ARTIFICIAL     SPACE-RELATIONS. 

(^Edcperiment  H.  —  Preliminary  Report!) 

§  71.  Having  conducted  the  following  experiment 
upon  only  one  person,  the  results  are  not  yet  well 
enough  determined  for  full  publication.  Yet,  their 
nature  is  already  such  as  to  warrant  mention  of  them 
among  our  other  experiments,  to  which  they  stand  in 
close  relation,  and  which,  from  the  first,  they  were 
designed  to  supplement. 

§  72.  The  apparatus  may  be  described  roughly,  as 
follows  :  Conceive  49  cylindrical  keys,  each  5  mm.  in 
diameter,  arranged  in  a  square,  7  keys  on  a  side.  The 
keys  are  kept  raised  by  springs,  and  when  one  is  pressed 
down,  a  pin  or  needle,  screwed  in  the  lower  end  of  the 
cylinder,  is  pressed  against  the  skin.  The  distance 
from  center  to  center  of  each  pin  is  1  cm.,  measured 
in  lines  parallel  to  the  sides  of  the  square.  The  whole 
is  secured  in  a  proper  box  or  frame.  The  peculiarity 
about  the  instrument  concerns  the  one  cylinder  in  the 
center  of  the  square.  This  has  no  pin  in  its  end,  but  is 
attached  to  a  lever  4  cm.  long,  which  runs  1  cm.  beyond 
the  outer  limits  of  the  square  formed  by  the  49  keys. 
In  the  end  of  this  lever  is  a  pin,  which,  when  the 
center  key  is  pressed  down,  is  forced  against  the  skin, 
similarly  as  are  the  other  pins.    The  center  key,  working 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       173 

the  lever,  must  be  so  adjusted  that  it  shall  feel  to  the 
subject  "  to  press  down  "  in  a  manner  not  noticeably 
different  from  the  other  keys  ;  the  crucial  thing  being 
that  he  shall  not  suspect  that  the  pin  it  presses  on  the 
skin  is  not  in  the  end  of  the  cylinder  directly  under  its 
center,  precisely  as  are  all  the  other  pins.  To  this 
end,  also,  the  box  is  made  so  as  to  conceal  its  inner 
arrangement.  The  top  shows  only  the  heads  of  the 
keys  numbered  with  plain  figures  in  regular  order.  The 
bottom  is  a  perforated  board,  on  a  level  just  below 
the  end  of  the  pins  as  they  rest  without  being  pressed 
down.  The  ends  of  the  pins  are,  therefore,  not  seen  if 
the  box  be  raised  and  examined  ;  and  as  the  bottom  is 
perforated  all  over,  and  not  alone  for  the  pins  just  in 
the  square,  the  subject,  with  proper  care,  is  little  likely 
to  suspect  the  one  thing  that  must  be  kept  from  him, 
namely,  the  true  position  of  the  "  decoy  "  pin  worked 
by  the  center  key.  The  box  is  carried  by  an  adjustable 
arm  and  stand  that  permits  it  to  be  lowered  and  held 
over  any  region  of  skin,  as  near  as  may  be  desired, 
without  resting  upon  it. 

Various  "■  dies  "  are  now  made  by  cutting  holes  out  of 
heavy  pasteboard  squares,  or  notches  out  of  strips,  to 
the  end  that,  by  use  of  them,  any  desired  combination 
of  keys  may  be  pressed  down  at  once. 

§  73.  The  intention  of  the  experiment  is  to  cultivate 
artificial  space-relations  by   continued   education   upon 


174       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

regions  of  the  skin,  which  have  a  high  degree  of 
native  sensitiveness  or  capability  of  dermal  discrimi- 
nation, but  which,  from  reasons  of  their  location  and 
"  lot "  in  life,  have  received  but  very  meager  spatial 
education  and  perception  development.  The  notion  is 
that,  in  such  regions,  one's  natural  spatial  discrimina- 
tion will  at  first  be  unable  to  distinguish  the  true  posi- 
tion of  the  decoy  pin ;  its  education  will  depend  upon 
the  actual  serial  relations  artificially  formulated  by  the 
experiments  ;  in  these  series  the  decoy  pin  is  made  to 
fall  into  the  same  time  series  that  it  naturally  would 
fall  into  were  it  situated  in  the  center  of  the  square 
instead  of  outside  of  it.  Consequently,  to  the  subject, 
the  feeling  of  this  pin  gets  woven  into  his  newly- 
educated  space-nature  in  precisely  the  same  relations  as 
would  have  happened  were  the  decoy  pin  really  in  the 
middle  of  the  square,  where  the  subject  from  the  first 
has  supposed  it  to  be.  , 

We  selected  for  our  first  experiment  the  abdomen, 
which,  besides  filling  the  above  requirements,  enabled 
the  subject  to  see  and  operate  the  apparatus.  The 
experiment  was  continued  ten  or  fifteen  minutes  for 
nearly  three  months.  I  explained  to  my  subject  that 
my  purpose  was  to  test  the  limit  to  which  one's  power 
of  dermal  perception  or  discrimination  could  be  edu- 
cated. The  plan  was  to  begin  with  so  many  keys  or 
pins  that  the  subject  could  not  possibly  discriminate  the 


OUR    NOTIONS    OF    NITMBER   AND    SPACE.       175 

true  position  of  the  decoy  pin,  or,  in  fact,  of  any  other 
single  pin.  It  will  be  understood  that  this  was  not 
difficult  to  do,  since  the  distance  apart,  here,  at  which 
two  compass  points  can  be  distinguished  separately 
is,  according  to  Weber,  from  5  to  7  cm.  We  then  pro- 
ceeded to  drop  out  certain  pins,  or  rows  of  pins,  always 
working  toward  distincter  masses  and  combinations. 
At  the  end  of  a  week  I  began  using  full  rows  of  pins, 
never  fewer,  at  that  time.  The  subject  would  work  a 
while,  pressing  the  keys  himself,  and  trying  to  fasten  in 
his  mind  just  how  the  several  rows  individually  felt. 
He  would  then  close  his  eyes  and  I  would  press  the 
keys  of  one  row  and  then  another,  asking  him  each 
time  to  identify  the  row  he  felt. 

At  the  end  of  two  weeks  I  began  dropping  out  pins 
from  the  single  rows,  and  combinations  of  pins  in  the 
same  row.  This  was  kept  up  till  the  summer  vacation 
began,  when  we  were  compelled  to  suspend  the  experi- 
ment. But  such  progress  had  been  made  that  finally  I 
was  able  to  press  successively,  in  any  order,  the  two  end 
and  the  middle  (our  decoy)  pin  in  any  row,  and  the 
subject  correctly  locate  the  keys  that  had  been  pressed, 
and  the  order  in  which  he  had  felt  them ;  yet  do  so 
without  the  least  suspicion  that  the  middle  or  decoy  pin 
was  outside  of  the  square  altogether,  and  3  cm.  distant 
from  where  he  felt  it  to  be,  namely,  right  under  the 
center  of  the  key  that  pressed  it  down.     At  times,  also, 


176       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

I  made  tests  like  the  following :  Pressing  one  end  key 
and  the  middle  key  of  any  row,  I  would  then  press  the 
other  end  key  of  the  same  row,  and  ask  if  the  last  pin 
was  ''between,"  or  outside  the  two  former  ones.  Or, 
again  press  the  two  end  keys  and  then  the  middle  key, 
and  ask  the  same  question.  Though  the  subject  would 
not  always  give  the  correct  answer,  he  would  do  so  in 
more  than  80  per  cent  of  the  cases,  which  is,  perhaps, 
as  much  as  he  would  have  done  had  the  decoy  pin 
really  been  in  the  center  where  he  conceived  it  to  be. 

The  significance  of  this  experiment  in  the  light  of 
the  doctrine  of  the  Genetic  origin  of  our  spatial  per- 
ception, is  obvious.  Artificially,  we  seem  to  have 
actually  demonstrated  here  what  the  Genetic  Theory 
asserts  theoretically  :  that  the  absolute  fixity  which 
appears  to  characterize  the  spatial  relations  of  one  point 
to  another,  and  of  one  line  and  surface  to  another  in 
those  objective  perceptions  of  places  and  things  which 
we  call  "reaZ,"  is  wholly  dependent  upon  a  certain 
definite  fixity  of  time-order  or  "time-relatio7i"  in  our 
original  experiences,  and  which  is  preserved,  with 
greater  or  less  integrity,  in  our  habits  of  mental  repro- 
duction. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       177 

GENERAL    SURVEY    AND    SUMMARY. 

We  have  now  examined  the  results  of  our  several 
experiments,  and  discussed  many  questions.  Other 
points  of  importance  remain  to  be  considered  under 
a  comparative  survey  of  the  several  tables.  But  I 
must  be  content  to  knit  these  up  with  a  general 
survey  of  the  whole  field.  I  will,  therefore,  close 
the  present  paper  with  a  summary  of  our  thesis  as 
it  now  stands  : 

1.  Certain  parts  of  the  brain  react  correspond- 
ingly to  certain  definite  combinations  of  peripheral 
nerves. 

2.  Every  combination  (peripheral  or  central)  acting 
together  once,  develops  to  that  degree  a  tendency  to 
act  together  again ;  the  more  frequently  the  parts  act 
together  the  stronger  is  this  tendency. 

3.  This  law  holds  good,  within  certain  limits,  as 
well  of  successive  combinations  or  series,  as  of  simul- 
taneous combinations. 

4.  Successive  combination  influences,  in  a  marvelous 
and  wholly  unexplainable  manner,  the  simultaneous 
activities  of  the  same  parts. 

5.  Until  successive  stimulation  of  the  combination 
occurs,  the  mental  correspondent  of  the  simultaneous 
stimulation  of  it  will  be  one  homogeneous  qualitative 
whole. 


178       OUR   NOTIONS    OF    NUMBER    AND    SPACE. 

6.  Subsequent  to  proper  stimulation  of  the  total 
combination  successively  in  definite  parts,  the  mental 
result  of  the  simultaneous  stimulation  of  the  whole 
■will  present  a  collection  of  parts  so  arranged  that  the 
successive  qualitative  wholes  which  followed  each  other 
in  definite  order  in  the  original  succession,  will  be 
represented  in  a  simultaneous  picture  or  presentation  of 
them,  by  corresponding  qualitative  parts. 

7.  In  the  simultaneous  presentation  each  infinitesimal 
part  bears  a  qualitative  nature  that  can,  by  the  psychol- 
ogist, be  traced  back  to  a  greater  or  less  dependence 
upon  the  quality  of  a  certain  corresponding  term  in  the 
original  succession. 

8.  Also  all  these  parts  are  arranged  in  a  manner 
respectively  expressive  of  the  positions  which  their 
corresponding  terms  occupied  in  the  time-order  of  the 
original  succession. 

9.  If  the  terms  of  the  original  succession  weye  suffi- 
ciently broken  in  order,  or  unlike  in  quality,  a  numerical 
presentation  will  result  from  the  subsequent  simul- 
taneous stimulation. 

10.  If  the  original  series  was  continuous  a  distance 
presentation  is  given  —  as  a  line. 

11.  If  the  original  was  continuous,  but  its  terms 
unlike  qualitatively,  the  presentation  will  be  both 
numerical  and  spatial;  as  a  line  of  bead-like  intensities, 
or  of  different  colors. 


OUR    NOTIONS    OF    NUINIBER   AND    SPACE.       179 

12.  Dermal  presentations  are  chiefly  founded  upon 
original  terms  alike  in  kind  (not  different  as  are  colors) 
but  differing  in  intensities. 

13.  Numerical  and  distance  presentations  are  not 
governed  alone  by  tlie  last  preceding  successive  stimu- 
lations of  their  corresponding  sets  of  peripheral  nerves, 
but, 

14.  Since  all  the  past  serial  modes  in  some  degree 
modify  the  simultaneous  reac^on,  and  this  at  any  one 
time  can  present  but  one  serial  arrangement,  the  latter 
becomes  an  averaged  resultant  of  all  the  serial  modes 
which  the  given  peripheral  combination  has  experienced 
throughout  life. 

15.  If  any  peripheral  combination  through  life  has 
been  stimulated  sufficiently  more  together  than  sepa- 
rately, the  presentation  will  be  a  numerical  ''  Avhole." 
As  pin-point  and  small  dermal  areas. 

IG.  If  the  nerves  are  strung  out  in  a  line,  and  the 
line  is  stimulated  through  life  more  serially  than  all 
together,  the  presentation  will  be  a  lineal  "■  distance 
presentation." 

17.  The  distance  presentation,  or  the  apparent 
length  of  the  line  will  express  the  average  length  of  all 
the  time  series  in  which  the  peripheral  line  has  through 
life  been  stimidated. 

18.  Upon  simultaneous  stimulation  of  any  dermal 
surface,   there  tend  to   rise  into  presentation  simulta- 


180       OUR    NOTIONS    OF   NUMBER    AND    SPACE. 

ueously,  all  the  distance  series  developed  between  every 
possible  pair  of  points  in  that  surface ;  the  consequent 
presentation  is  a  specific  resultant  of  the  sum  of  all 
those  particular  tendencies ;  such  we  call  a  "  surface  " 
presentation. 

19.  Since  each  separate  tendency  that  is  active  at 
any  moment  modifies  and  is  modified  by  every  other 
tendency  active  at  that  moment,  the  apparent  distances, 
in  the  resultant  presentation,  are  modified  according  to 
the  particular  peripheral  arrangements  combined  in 
stimulation  at  that  moment. 

20.  All  distance  and  spatial  perceptions  are  partly 
dependent  upon  the  particular  peripheral  arrangements 
to  whose  simultaneous  stimulations  they  respond ;  and 

21.  Partly  upon  the  average  sort  of  distance-series 
experienced  through  life  between  each  pair  of  points  in 
these  arrangements. 

22.  The  time-length  of  the  serial  stimulal^ions,  re- 
ceived on  any  particular  region  of  the  body,  will  depend 
partly  upon  the  contour  of  that  region  ;  and 

23.  Partly  on  the  outer  events  to  which  that  region 
is  customarily  subjected. 

24.  Lines  two  feet  long  can  be  drawn  on  the  leg; 
such  cannot  be  drawn  on  the  tongue.  Every  nerve  in 
any  region  is  likely  to  enter  into  every  length  of  com- 
bination possible  to  that  region.  Every  length  of  com- 
bination has  proportionate  influence  on  all  subsequent 


ouii  NOTIONS  OF  nu:mber  and  space.     181 

tendencies  in  which  that  nerve,  by  life's  deyelopments, 
is  involved.  Consequently,  all  the  distance  presenta- 
tions of  the  leg  are  likely  to  appear  relatively  longer 
than  those  of  the  tongue,  or  of  other  small  members. 
On  the  other  hand,  since  short  lines  can  be  drawn  on 
any  member,  large  or  small,  and,  since  any  line,  long  or 
short,  may  be  drawn  at  any  rate  of  motion,  quick  or 
slow,  therefore,  the  average  distance-tendencies  devel- 
oped for  an}^  particular  region  will  depend  chiefly  upon 
the  outer  events  of  tliat  region. 

25.  Partly,  the  distance  tendencies  will  depend  on 
the  length  of  lines  most  commonly  drawn  on  a  given 
region ;  and 

26.  Partly,  on  the  rates  of  motion  most  common  in 
the  drawing. 

27.  For  various  stretches  of  skin  not  too  distant 
from  each  other,  or  covering  mjembers  not  too  unlike  in 
function,  the  average  length  of  lines  drawn  are  likely  to 
be  approximately  the  same,  and  so,  also,  the  average 
time-rates  for  the  two  stretches.  Hence,  the  presenta- 
tions of  different  lengths  of  peripheral  lines  are  approx- 
imately proportional  to  their  absolute  lengths. 

28.  For  regions  distant  from  each  other  (as  fore- 
head and  abdomen),  or  covering  members  of  unlike 
function  (as  tongue  and  cheek),  the  lines  drawn, 
and  the  rates,  are  likely  to  average  unequally.  Hence, 
the  respective   presentations   of  equal  peripheral  lines 


182       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

upon    tlie  two    regions,  commonly  appear  of  different 
lengths. 

From  all  of  the  above,  the  following  laws  may  be 
deduced,  relative  to  different  regions. 

29.  The  larger  the  region,  the  longer  will  its  units 
of  peripheral  distance  appear  to  be  in  spatial  presenta- 
tions. 

30.  On  any  region  longer  in  one  direction  than  in 
another,  the  units  of  the  longer  direction  will  appear 
longer  than  those  of  the  other. 

31.  The  longer  the  motions  commonly  made  by  or 
on  any  member,  the  longer  will  its  presentations 
appear.  (Those  of  forearm  are  longer,  relatively,  than 
those  of  upper-arm.) 

32.  Same  for  direction.  (Thus,  forearm  makes  wider 
sweeps  radially  than  forward  and  backward,  like  a 
piston.  As  our  experiments  show  that  the  distances 
across  are  seemingly  greater  than  along  the  firearm, 
we  may  presume  that  the  advantage  of  wider  sweep 
overcomes,  here,  the  disadvantage  of  less  anatomical 
length.) 

33.  The  more  rapid  the  movements  made  by  or  on 
any  member,  the  shorter  its  presentations  appear. 

34.  Same  for  direction. 

35.  Gravity  being  a  force  continually  tending  to 
lengthen  movements  over  the  skin  in  a  downward  direc- 
tion, thus  making  the  average  downward  experiences  of 


OUR   NOTIONS    OF    NUJMBER   AND    SPACE.       183 

life  longer  than  others,  therefore,  vertical  presentations 
are  apt  to  appear  longer  than  horizontal  ones. 

36.  The  more  frequent  any  particular  length  or  rate 
of  moyement  on  any  region,  the  more  do  all  the  presen- 
tations of  that  region  approximate  to  that  length. 

37.  Same  for  direction. 

So  far,  I  have  discussed  only  continuous  lines  and 
stretches  of  skin.  I  will  now  consider  portions  of  skin 
not  continuous. 

38.  If  any  two  separate  points  be  simultaneously 
stimulated,  the  presentation  will  express  the  average  of 
all  serial  combinations  in  which  those  two  points, 
through  life,  have  participated.  (Same  as  Law  Four- 
teen, for  continuous  lines  or  areas.) 

39.  Since  an  infinite  number  of  curved  lines  of 
varied  lengtli  can  and,  through  life,  are  likely  to  be 
drawn  between  every  pair  of  points,  therefore  they 
become  knit  up  into  an  infinite  number  of  tendencies. 

40.  The  averages  of  these,  for  different  pairs  of 
points  on  similar  regions,  will  be,  respectively,  propor- 
tional to  the  corresponding  right-line  distances  between 
those  points. 

41.  Laws  Six,  Seven  and  Eight  hold  good  under  the 
supposition  that  the  present  stimulation  affects  all 
terms  of  the  peripheral  combination  with  equal  inten- 
sity. If  it  affects  some  terms  with  greater  intensity 
than  others,  corresponding  effects  of  intensity  will 
appear  in  the  proper  terms  of  the  presentation. 


184       OUR    NOTIONS    OF   NUMBER    AND    SPACE. 

42.  Consequently,  in  presentations  from  separated 
points,  the  terms  correspondent  to  those  points  appear 
with  increased  intensity. 

43.  The  more  frequently  any  given  separated  pres- 
entation is  awakened  by  proper  peripheral  stimulation, 
the  more  marked,  through  the  modifying  influence  of 
each  repetition,  becomes  the  difference  of  intensity 
between  the  correspondent,  or  stimulated  terms  and  the 
remaining  terms,  i.e.,  the  more  distinct  and  differenti- 
ated becomes  the  presentation. 

I  esteem  Laws  Forty-one,  Forty-two  and  Forty-three 
to  be  of  the  highest  importance  in  all  psychological 
science.  From  them,  chiefly,  arise  the  highly  differen- 
tiated and  clear  mental  states  which  we  call  perceptions 
of  definite  objects  and  things.  They  largely  account  for 
the  differences  of  clearness,  strength  and  coherency 
between  sensations  and  ideas  ;  between  perceptions  and 
conceptions.  » 

44.  The  more  distinct  and  differentiated  the  sepa- 
rate terms  of  a  presentation  become,  the  more  distinctly 
do  they  become  numerical  presentations. 

45.  To  formulate  the  genesis  of  numerical  presenta- 
tions, we  must,  therefore,  determine  the  laws  governing 
the  simultaneous  and  successive  combinations  of  sepa- 
rate points.     This  may  be  done  as  follows  : 

46.  The  further  points  are  apart,  the  more  likely 
they  are  to  be  stimulated  separately. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       185 

47.  The  more  they  are  stimulated  separately,  the 
more  distinct  and  accurate  are  their  corresponding 
numerical    percepts. 

48.  The  further  points  are  apart,  the  more  distinct 
and  accurate  are  their  numerical  percepts.  (This  is  our 
old  Law  Three.) 

49.  The  more  frequently  any  x  number  of  separate 
points  are  stimulated  in  the  same  combination,  succes- 
sively or  together,  tbe  more  distinctly  and  accurately 
will  these  presentations  be  of  x  numbers. 

50.  In  any  fixed  area  of  skin,  the  greater  the  number 
of  points  in  any  given  combination,  the  less  frequently 
through  life  are  they  likely  to  be  stimulated  together, 
either  simultaneously  or  successively. 

51.  Therefore,  the  greater  the  number  of  points  in 
any  given  distance,  the  less  accurate  and  clear  those 
numerical  perceptions. 

52.  Or,  for  any  given  category  of  number,  the 
greater  the  distance,  the  more  certain  and  clear  the 
numerical  perceptions.  (Our  old  Law  Two,  for 
Number.) 

53.  Since  stinmlation  of  two  separate  peripheral 
points,  A  B,  tends  to  recall  the  distance  tendency 
corresponding  to  the  right-line,  AB ;  and  since  the 
stimulation  of  any  other  points  on  the  line,  AB,  simul- 
taneously with  the  stimulation  of  A  B,  tends,  also,  to. 
recall  this  same  right-line  tendency;  therefore,  ojar^y 


186       OUR    NOTIONS    OF    NUMBER    AND    SFACE. 

given  peripheral  right-line,  the  greater  the  number  of 
separate  points  stimulated  simultaneously,  the  more 
accurately  will  the  right-line  presentations  arise — (com- 
monly we  would  say,  the  more  accurately  will  it  be 
judged). 

54.  Since  the  right-line  presentation  is  the  shortest 
presentation,  the  greater  the  number  of  points,  the 
shorter  will  the  line  appear.  (Our  old  Law  Four,  for 
Distance.) 

55.  Of  the  presentations  of  the  same  absolute  dis- 
tance, those  from  a  continuous  straight-edge  will  appear 
shorter  than  those  from  separated  points.  (This  law  is 
for  simultaneous  stimulation.  The  reverse  may  hold 
good  for  successive.) 

56.  Other  things  being  equal,  the  further  apart  two 
points  are,  the  less  likely  is  a  line  to  be  drawn  between 
them. 

57.  Since  the  accuracy  of  a  distance  presentation 
depends  on  the  frequency  of  its  proper  stimulation, 
therefore,  from  the  above,  other  things  equal,  the 
greater  the  distance,  the  less  accurate  shovdd  be  its 
presentation. 

58.  On  the  other  hand,  since  the  simultaneous 
stimulation  of  separate  points  calls  up  the  straight-line 
tendency  developed  between  them,  and,  by  so  doing, 
strengthens  that  tendency;  therefore,  other  things 
being  equal,  the  greater  the  distance,  the  more  accurate 


OUR   NOTIONS    OF    NUMBER    AND    SPACE.       187 

should  be  its  presentation.  This  is  Init  saying;  The 
more  distinctly  separated  our  notions  of  local  points 
are,  the  more  accurate  our  distance-judgments  of  them. 
(Our  old  Law  Two,  for  Distance.) 

59.  By  reason  of  the  conflict  of  above  Laws  Forty- 
two  and  Forty-three,  and  because  many  varied  local 
conditions,  formulated  by  our  several  above  laws 
together,  govern  the  sort  of  distance-series  most  com- 
mon to  each  particular  region ;  therefore,  the  absolute 
distance  most  correctly  judged,  for  each  particular 
region,  must  be  determined  empirically,  and,  when  so 
determined,  is  the  resultant  expression  of  all  these 
conditions. 

60.  When  so  determined  for  any  given  region,  it  will 
hold  good  for  that  region,  that  all  distances,  longer  or 
shorter  than  this  preferable  distance,  will  be  presented 
with  proportionally  less  accuracy  than  the  preferable 
distance.     Consequently  — 

61.  For  distances  above  the  preferable  distance,  the 
greater  the  distance  the  less  accurate  the  presentations. 
And— 

62.  For  distances  below  the  preferable  distance,  the 
greater  the  distance  the  more  accurate  the  presenta- 
tions. (This  goes  with  our  Law  Two  of  the  Distance- 
judgments  in  our  various  experiments.) 

63.  All  the  above  laws,  both  of  Xumber  and  of 
Distance,  hold  good  in  their  general  intent,  —  i.e.,  in 


188      OUR   NOTIONS   OF   NUMBER   AND   SPACE. 

SO  far  as  the  actual  result  is  not  modified  by  new 
conditions — in  the  more  complex  presentations  of  two- 
dimensioned  space. 

64.  Since  each  distance  tendency  awakened,  at  any 
given  time,  tends,  in  its  measure,  to  modify  all  the  other 
distance  tendencies  active  at  that  time  ;  therefore,  if  we 
attempt  to  judge  any  particular  element  or  line  of 
distance,  in  any  given  peripheral  combination,  the  con- 
sequent percept,  or  judgment,  will  be,  in  some  degree, 
modified  from  what  it  would  be,  were  that  element 
stimulated  by  itself,  and  will  be  modified  proportionally 
to  the  mean  of  all  the  distance  tendencies  developed 
between  every  possible  pair  of  points  in  the  total 
peripheral  combination.  This  I  will  hereafter  speak  of 
as  the  Law  of  Average  Distances. 

65.  Hence,  of  the  distance-judgments  of  lineal  fig- 
ures, those  of  equilateral  triangles  are  less  than  those  of 
circles  ;  those  of  circles  less  than  those  of  squaras  ;  and 
those  of  squares  less  than  judgments  of  single  lines 
presented  independently  of  all  figured  implications. 
(The  basis  of  comparison  is  assumed  here  to  be  the 
sides  of  the  right-line  figures  and  the  diameters  of  the 
circles.) 

66.  For  similar  reasons,  the  greater  the  number  of 
internal  points,  in  any  figured  arrangement  of  points, 
the  shorter  will  be  the  distance  presentations.  (Those 
of   the   four-pin   triangle,   of  our  Experiment    B,   will 


OUR    NOTIONS    OF    NUMBER    AND    SrACE.       189 

appear  shorter  than   those    of  the   three-pin   triangle, 
etc.) 

67.  Since  (Law  Fifty-two)  the  greater  the  dis- 
tance, the  more  accurate  the  numerical  presentations, 
and  since,  in  two-dimension  figures,  the  distance  ten- 
dencies modify  each  other  proportionally  to  the  aver- 
age distances ;  therefore,  the  numerical  presentations 
of  separate  points,  set  in  figured  arrangements,  will  be 
accurate  and  clear  in  proportion  to  the  average  dis- 
tance of  all  the  points.  (Four  points  in  a  square  will 
be  more  accurately  presented  than  four  in  a  triangle,  or 
four  in  a  line,  etc.) 

68.  Owing  to  the  facts  —  {a)  That  different  parts  of 
the  brain  act  on  different  occasions  ;  (J))  That  parts  which 
have  acted  together  serially  tend  to  repeat  these  series  ; 
(c)  That  by  varied  peripheral  experiences,  the  several 
central  parts  have  been  combined  into  innumerable 
different  series  ;  (d)  That  every  passing  presentation 
being  vastly  complex,  the  sum  of  tendencies  of  which 
it  is  the  resultant  must  at  all  times  contain  terms  which 
are  members  of  some  certain  series  having  yet  remaining 
terms  uncompleted  in  their  passing,  and,  therefore,  at 
the  given  moment  having  yet  a  certain  remnant  of 
tendency  toward  continuance ;  (e)  And  that  proper 
nourishing  processes  renew  the  parts  to  approximately 
constant  inclination  to  activity  ;  therefore,  the  vast 
complexity   of    centrally   developed   serial    tendencies 


190       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

once  under  way,  they  continue  to  combine  and  to  recom- 
bine  indefinitely  and  independently  of  freshly  incoming 
peripheral  impulses. 

69.  The  content  of  thought  at  any  given  moment 
expresses  the  sum  of  two  separate  sources  of  influence  : 
the  already  progressing  central  processes  and  the 
incoming  peripheral  influences. 

70.  Both  processes  play  ultimately  upon  the  same 
brain  parts  and  upon  the  same  general  storehouse  of 
developed  tendencies.  The  tendencies  that  at  one 
moment  are  active  in  the  central  stream,  are  identically 
the  same  that  at  another  moment,  or  it  may  be  at  the 
same  moment,  are  played  upon  by  the  incoming  periph- 
eral stream  ;  those  of  both  are  of  the  same  general 
origin  and  nature,  and  governed  by  the  same  laws. 
Therefore  :  all  the  foregoing  laws  of  combination  and 
development,  apply  as  well  to  the  central  stream  as  to 
the  peripheral  stream,  and  as  well  to  the  joint  combina- 
tion of  the  two  streams  as  to  either. 

71.  Every  passing  presentation  expresses  the  re- 
sultant of  the  sum  of  the  joint  streams  of  tendencies  — 
the  peripheral  and  the  central  — under  mutual  modifica- 
tion of  education  ;  those  of  the  one  working  to  modify 
those  of  the  other  equally  with  those  of  its  own. 

72.  According  to  the  methods  of  judging  peripheral 
combinations  used  in  the  experiments  of  this  paper,  the 
subject   knew   the   certain  range   of   presentation   and 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       191 

categories  which  he  coukl  expect.  Consequently,  from 
the  expecting  of  these,  —  from  the  ranging  of  the 
imagination  over  the  possible  categories  to  be  applied,  — 
a  certain  range  of  tendencies  corresponding  to  the 
range  of  experimentation  were  already  awakened  into 
activity,  preparatory  to  the  application  of  the  peripheral 
category. 

73.  Also,  the  concentration  of  the  attention  upon  a 
particular  region  awakens,  in  a  vaguer  degree,  all  the 
categories  common  to  that  region. 

74.  Consequently,  upon  application  of  any  single 
piece  of  apparatus,  three  distinct  sets  of  categories 
were  awakened,  from  the  joint  action  of  which  the 
final  perception  was  an  expression ;  namely  (1)  The 
tendencies  developed  for  the  special  peripheral  combina- 
tion actually  applied.  (2)  The  range  of  tendencies 
expected  and  governed  by  the  range  of  experiment. 
(3)  The  possible  range  of  tendencies  developed  for  this 
particular  region. 

75.  The  presentation  resulting  from  these  three  sets 
of  tendencies  will  be  governed,  first,  proportionally  by 
their  respective  strengths  —  and  second, 

76.  Proportionally  by  their  mutual  congruity. 

77.  Of  the  three  sets,  the  influence  of  the  peripheral 
tendencies  is  likely  to  be  the  strongest,  those  of  the 
expected  categories  the  next  strongest,  and  the  local 
possible  categories  the  weakest ;  and  for  the  following 


192       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

reasons  :  The  application  of  the  apparatus  by  external 
means  continues  the  influence  of  one  single  definite 
combination  unchanged  through  a  relatively  prolonged 
time.  On  the  other  hand,  owing  to  the  continued 
"  ranging "  of  the  expectations,  the  influence  of  each 
particular  expected  category  is  less  sustained,  and  also 
by  the  modification  of  mutual  successions,  becomes 
weaker  and  less  explicit.  The  possible  categories  are 
even  more  flittingly  and  vaguely  awakened.  Moreover, 
the  periplieral  tendencies  form  a  definite  combination 
ill  the  habit  of  liolding  together  for  a  prolonged  time, 
while  the  expected  categories  not  only  do  shift  and 
range  in  perpetually  changing  combinations,  but  never 
have  had  ani/  fixed  order  of  combination,  such  as  is 
prescribed  to  them  by  the  range  of  the  experiment  and 
by  the  vast  variety  of  the  passing  stream  of  present 
events. 

78.  Therefore  :  the  consequent  presentation  is  likely 
to  be  moulded  most  by  the  form  of  the  peripheral  com- 
bination ;  next  in  amount  by  the  expected  range  of 
categories  set  by  the  experiment  ;  and  least  by  the 
possible  categories  set  by  the  character  of  the  particular 
region  worked  on. 

79.  The  fewer  times  any  given  peripheral  combina- 
tion has  occurred,  the  less  strong  will  it  be,  and  the 
more  will  the  consequent  presentation  express  the 
average  of  the  several  other  combinations  into  which, 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       193 

through  life,  the  separate  terms  of  the  given  combina- 
tion have  severally  entered. 

80.  Therefore  :  the  weaker  or  less  developed  the 
given  peripheral  combination  on  any  region,  the  more 
will  the  presentation  drift  toward  an  expression  of  the 
average  of  all  the  possible  categories  of  that  particular 
region.     (First  form  of  our  Law  Three.) 

81.  If  the  applied  category  is  below  the  average  of 
total  categories  developed  for  the  given  region,  the 
drift  of  error  will  be  toward  "  over-estimation "  ;  if 
above,  toward  under-estimation. 

82.  Since  the  weaker  the  peripheral  influence,  the 
stronger  proportionally  is  the  central  influence,  there- 
fore :  the  less  developed  or  ''  more  uncertain "  the 
peripheral  combination,  the  more  will  the  presentation 
drift  to  an  average  of  the  expected  categories. 

83.  Since  the  shorter  the  distance  the  more  "  uncer- 
tain "  the  presentations,  therefore  :  the  shorter  the 
distance  the  greater  the  over-estimation,  both  by  reason 
of  81  and  of  82.  (This  is  our  old  Law  Three,  of 
Number  and  of  Distance.) 

84.  From  81,  therefore :  in  general,  small  numbers 
and  sliort  distances  are  over-estimated  ;  large  numbers 
and  long  distances,  under-estimated. 

85.  Since  we  call  those  presentations  the  most 
"  accurate  "  which  we  receive  through  our  highest  de- 
veloped  region   or   sense,  and   since  our  several  dermal 


194       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

regions  are  variously,  and.  some  of  them  very  inferiorly, 
developed;  therefore  :  for  many  regions  what  are  the 
most  constant  and  certain  judgments  for  that  region 
may  not  be  what  we  commonly  conceive  of  as  the  most 
"accurate"  judgments,  even  from  that  region.     Also, 

86.  Since  the  drift  due  to  the  inaccuracy  from  the 
category  actually  applied  to  a  given  region  may  com- 
pensate the  whole  general  drift  of  the  region  in 
comparison  with  the  judgments  of  our  highest  developed 
senses,  therefore:  we  often  call  judgments  "accurate" 
which  are  functionally  very  inaccurate  for  the  region 
and  categories  really  involved.  (By  these  compensations 
are  explained  the  phenomenon,  so  common  through- 
out all  our  experiments,  of  apparently  increasing 
accuracy,  through  those  categories,  where,  by  all  our 
most  general  laws,  there  ought  to  be  displayed  an  ever- 
increasing  inaccuracy.  As  for  instance,  in  all  the 
shorter  distance  categories  of  Experiment  A,  where  we 
discovered  constantly  increasing  numbers  of  correct 
judgments  with  increasing  numbers  of  pins,  although 
really  the  greater  the  number  of  pins  the  more  difficult 
and  uncertain  the  dermal  presentations  ought  to  be.) 

87.  Since  the  different  regions  and  members  of  our 
body  vary  greatly  in  contour,  size,  function  and  lot  in 
life's  experience,  therefore  and  consequent  to  the  above: 
the  whole  general  scale  of  presentation  for  different 
regions  and  members  vary  greatly  one  from  another. 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       195 

88.  Our  most  mobile  regions  are  the  most  used  ; 
therefore,  the  most  highly  developed  ;  therefore,  their 
scales  of  judgments  are  deemed  the  most  accurate. 

89.  As  a  matter  of  fact,  our  highest  developed 
regions  are  not  only  the  most  mobile,  but  also  are  both 
our  smallest  and  our  quickest-moving  members  —  (eye, 
in  proportion  to  number  of  nerves  stimulated,  tongue 
and  lingers). 

90.  Hence,  our  most  accurate  scales  of  judgment  are 
"  short  scales  "  of  judgment,  relatively  to  the  scales  of 
our  larger  and  slower  moving  regions  and  members. 

91.  Hence,  so-called  "  over-estimation  "  is  the  com- 
mon trait  of  all  our  larger,  clumsier  and  less  developed 
dermal  regions.  (Hence,  the  over-estimations  common 
to  all  the  tables,  and  in  our  several  experiments  for  the 
various  regions  worked  upon,  the  eye  outscaling  them 
all.) 

92.  Hence,  also,  the  relative  order  of  accuracy, 
maintained  alike  throughout  all  the  several  kinds  of 
judgments,  Avhether  of  number,  of  distance,  of  figure 
or  otherwise,  which  is  observed  between  the  different 
regions  of  the  body.  (Shown  in  "  Amounts  of  Average 
Error,"  "  Number  of  Correct  Judgments,"  "  Over- 
estimation,"  etc.,  of  our  tables,  and  confirming  the 
following  order  for  the  parts  we  worked  on  :  Tongue, 
Forehead,  Forearm,  Abdomen.) 

93.  Hence,  also,  the  relative  accuracy  of  judgments 


196       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

made   in   different  directions   upon   the    same    general 
region. 

94.  Not  only  do  the  foregoing  laws  hold  good  for 
the  judgments  of  simultaneous  presentations,  but  also 
they  hold,  within  limits,  for  judgments  of  serial  stimu- 
lations (such  as  from  a  pencil  drawn  on  the  skin). 

95.  Therefore,  and  since  every  distance  presentation 
is  in  essence  an  expression  of  serial  time-form :  The 
particular  time-rate  of  movement  over  a  given  periph- 
eral distance  will  express  itself  in  the  resultant  pres- 
entations. 

96.  Hence,  quick  movements  seem  shorter,  and  slow 
movements  seem  longer. 

97.  Hence,  and  because  heavy  peripheral  pressure 
spreads  to  affect  actually  longer  stretches  of  skin,  and 
perhaps,  as  well,  because  the  greater  intensity  of  the 
resulting  nervous  currents  may  cause  the  central  re- 
action to  endure  longer,  or  to  spread  further  through 
the  central  parts,  therefore  :  The  heft  of  moving  periph- 
eral stimulations  governs,  in  some  degree,  the  resultant 
presentations. 

98.  Hence,  heavy  movements  seem  longer,  and  light 
ones  shorter. 

99.  The  foregoing  laws  also  manifest  other  modifica- 
tions under  more  complicated  modes  of  procedure 
appropriate  to  the  differences  in  method  involved.  As 
for  instance : 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       197 

100.  Where,  in  applying  certain  lines  and  figures, 
"  rocking  "  is  allowed,  in  imitation  and  revival  of  the 
original  successive  events  in  which  our  notion  of  such 
figures  originate,  the  presentations  are  appropriately 
more  accurate  than  where  the  rocking  is  carefully 
excluded. 

101.  It  may  be  observed  throughout  our  several 
tables,  and  best  in  the  Summary  Tables,  that  the 
accuracy  of  presentation,  as  shown  in  the  Xumber  of 
Correct  Judgments  and  in  the  Amounts  of  Error,  is 
relatively  greater  for  Distance  that  for  Number,  and 
for  Figure  than  for  Distance. 

102.  Also,  the  Xumber-judgments  vary  more  in 
accuracy,  as  between  the  several  regions  of  the  body, 
than  do  the  Distance-judgments,  and  the  Distance- 
judgments  than  do  the  Figure-judgments.  Our  thesis 
is  now  able  to  offer  explanations  of  these  things  as 
follows  : 

103.  The  several  terms  of  the  original  dermal  series, 
from  which  dermal  presentations  both  numerical  and 
spatial  develop,  are  of  identical  quality,  and,  to  a  very 
large  degree,  of  like  intensity  throughout  each  given 
event.  Consequently,  that  numerical  separateness, 
which  characterizes  our  topographical  notions  of  any 
dermal  region,  and  which  is  fundamentally  based  upon 
difference  in  intensity  between  the  terms  of  the  de- 
veloped "  memory  series  "  of  each  particular  region,  is 


198       OUR    NOTIONS    OF    NUMBER    AND    SPACE. 

almost  wholly  dependent,  for  its  origin  and  development, 
upon  the  laws  of  simultaneous  stimulation  of  separate 
points.  ^ — Stimulation  of  the  separate  points  calls  up  the 
distance  tendency  of  the  whole  right-line  of  intermediate 
points,  but  emphasizes  the  two  end  points  by  the  passing 
stimulation  of  them,  while  the  intermediate  points  are 
not  so  emphasized  ;  hence,  the  difference  of  intensity, 
which  is  transmitted  and  augmented  from  repetition  to 
repetition  of  the  particular  occurrence.  Hence,  it  will 
be  observed  that,  in  dermal  presentations,  the  numer- 
ical development  is  dependent  upon  previous  distance 
development ;  consequently,  for  the  most  part  they  remain 
inferiorly  and  less  accurately  developed.  It  is  instructive 
to  note  that  in  hearing  the  conditions  are  wholly  different. 
The  difference  in  quality  of  the  many  tones  is  a  rich 
basis  in  itself  for  the  development  of  numerical  sepa- 
rateness  ;  and  there  is  little  if  any  development  based 
upon  the  spatial  arrangements  of  the  nerve-ends. ,  Con- 
sequently, our  auditory  numerical  judgments  are  highly 
developed  and  accurate,  while  we  are  capable  of  com- 
paratively little  strictly  auditory  spatial  discrimination. 
In  the  eye  we  have  rich,  qualitative  and  spatial  basis 
for  perception.  Consequently,  visual  perceptions  are 
highly  developed,  both  numerically  and  spatially. 

104.  The  numerical  range  of  possible  combinations 
on  any  given  region  is  practically  unlimited,  while  the 
distance  range  is  much  determined  by  the  local  contour 


OUR    NOTIONS    OF    NUMBER    AND    SPACE.       199 

and  dimensions  of  any  particular  region.  Any  number 
of  taps  can  be  made  successively  on  the  tongue,  but 
lines  more  than  two  or  three  inches  long  cannot  be 
drawn  on  the  tongue.  Consequently,  the  possibilities 
of  error  are  always  greater  for  numerical  than  for 
distance  presentation,  and  betray  themselves  more  under 
the  relatively  different  degrees  of  development  evolved 
between  the  several  regions  of  the  body. 

105.  Our  figure-judgments  (I  do  not  say  presenta- 
tions) are  based  Tjoth  upon  numerical  and  upon  distance 
presentations  ;  they  may  take  their  cue  from  either  and 
from  many  other  suggestions  in  operation  in  simpler  judg- 
ments but  too  complicated  for  the  limits  of  this  paper. 

106.  Hence,  judgments  of  certain  familiar  figures, 
as  of  the  various  triangles,  squares  and  circles  in  our 
experiments,  are  throughout  more  accurate  than  the 
judgments  of  the  very  numerical  and  distance  •presenta- 
tions upon  which  the  ^gUTe-judf/me7its  are  based. 

107.  Whether  we  judge  our  more  complex  figure 
presentations  as  wholes  —  as  when  judging  if  a  given 
impression  is  of  a  triangle  or  of  a  square — or  confine 
our  judgment  to  regard  some  particular  element  of  the 
total  presentation — as  when  judging  the  length  of  some 
one  side  of  a  triangle  —  the  results  give  evidence  of 
conformity  to  the  same  fundamental  laws  which  have 
governed  the  development  of  presentations  from  the 
beginning. 


200       OUR    NOTIONS    OF    NUIVIBER    AND    SPACE. 

108.  The  integrity  of  our  general  thesis  within  the 
realm  of  the  ideational  or  representative  processes,  as 
well  as  in  presentative  ones  more  directly  correspondent 
to  outer  impressions,  is  also  upheld  experimentally 
throughout  our  work. 

109.  Finally,  by  such  methods  as  that  of  P^xperi- 
ment  H,  we  are  able  artificially  to  construct  genuine 
inner  perceptions  of  space  and  spatial  relations  in  strict 
conformity  with  the  hypothetical  laws  which  we  have 
deduced  for  the  origin  and  development  of  all  spatial 
perceptions  naturally. 


Looking  backward,  we  may  now  summarize  our  thesis 
as  follows : 

Its  origin  and  foundation  must  be  fundamentally 
placed  in  the  following  law  :  Presentations  of  Number, 
of  Distance,  and  of  all  Spatial  Figures  and  ari-ange- 
ments  in  general,  are  alike  based,  primarily,  upon  serial 
events  differing  greatly  in  mode,  such  as  become  charac- 
teristic of  those  modes  of  presentation  which  we  call 
numerical,  extential  and  spatial,  but  all  of  them  gov- 
erned by  the  same  fundamental  laws  of  relationship. 
By  reason  of  this,  all  simultaneous  presentations  are 
dependent  upon,  and  expressive  of,  the  several  modes 
of  serial  occurrence  out  of  which,  through  life,  thexj 
have  evolved,  and  become  differentiated. 


OUE,   IS^OTIONS    OF    NUMBER    AND    SPACE.       201 

From  the  simplest  presentations  to  the  most  highly 
developed  functions  of  judgment,  we  find  this  same 
system  of  laws  articulated  everywhere  into  one  common 
Genetic  System  of  Mental  Development. 

Presenting  these  conclusions,  I  now  beg  the  reader 
to  turn  back  and  to  re-read  the  Introduction  of  this 
paper,  the  better  to  recall  its  full  purpose,  and  again  to 
orient  its  whole  subject  with  Psychological  Science  in 
general. 


^ 


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DUE  AS  STAMPED  BELOW 

OCT  12  1988 

;.U!ODi3Q.0CTO9  *Bi 

UNIVERSITY  OF  CALIFORNIA,  BERKELEY 
FORM  NO.  DD6                                BERKELEY,  CA  94720 

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OC.  BERKELEY L/BB/lB,Es 


C00S3733CJ5' 


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•'•/ 


